plosPLoS Comput BiolploscompPLoS Computational Biology1553-734X1553-7358Public Library of ScienceSan Francisco, USA08-PLCB-RA-0903R310.1371/journal.pcbi.1000328Research ArticleComputational Biology/Computational NeuroscienceNeuroscience/Sensory SystemsNeuroscience/Theoretical NeuroscienceEstimation of Thalamocortical and Intracortical Network Models from
Joint Thalamic Single-Electrode and Cortical Laminar-Electrode Recordings in the
Rat Barrel SystemNeural Network Model EstimationBlomquistPatrick1DevorAnna23IndahlUlf G.1UlbertIstvan45EinevollGaute T.1*DaleAnders M.3Department of Mathematical Sciences and Technology and Center for
Integrative Genetics, Norwegian University of Life Sciences, Ås,
NorwayAthinoula A. Martinos Center, Massachusetts General Hospital,
Charlestown, Massachusetts, United States of AmericaDepartments of Radiology and Neurosciences, University of California San
Diego, La Jolla, California, United States of AmericaInstitute for Psychology of the Hungarian Academy of Sciences, Budapest,
HungaryPeter Pazmany Catholic University, Department of Information Technology,
Budapest, HungaryFristonKarl J.EditorUniversity College London, United Kingdom* E-mail: Gaute.Einevoll@umb.no
Conceived and designed the experiments: AD IU GTE AMD. Performed the
experiments: AD IU. Analyzed the data: PB UGI GTE AMD. Contributed
reagents/materials/analysis tools: PB AD UGI GTE AMD. Wrote the paper: PB AD
UGI IU GTE AMD.
The authors have declared that no competing interests exist.
32009273200953e10003281010200810220092009Blomquist et alThis is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
A new method is presented for extraction of population firing-rate models for
both thalamocortical and intracortical signal transfer based on stimulus-evoked
data from simultaneous thalamic single-electrode and cortical recordings using
linear (laminar) multielectrodes in the rat barrel system. Time-dependent
population firing rates for granular (layer 4), supragranular (layer 2/3), and
infragranular (layer 5) populations in a barrel column and the thalamic
population in the homologous barreloid are extracted from the high-frequency
portion (multi-unit activity; MUA) of the recorded extracellular signals. These
extracted firing rates are in turn used to identify population firing-rate
models formulated as integral equations with exponentially decaying coupling
kernels, allowing for straightforward transformation to the more common
firing-rate formulation in terms of differential equations. Optimal model
structures and model parameters are identified by minimizing the deviation
between model firing rates and the experimentally extracted population firing
rates. For the thalamocortical transfer, the experimental data favor a model
with fast feedforward excitation from thalamus to the layer-4 laminar population
combined with a slower inhibitory process due to feedforward and/or recurrent
connections and mixed linear-parabolic activation functions. The extracted
firing rates of the various cortical laminar populations are found to exhibit
strong temporal correlations for the present experimental paradigm, and simple
feedforward population firing-rate models combined with linear or mixed
linear-parabolic activation function are found to provide excellent fits to the
data. The identified thalamocortical and intracortical network models are thus
found to be qualitatively very different. While the thalamocortical circuit is
optimally stimulated by rapid changes in the thalamic firing rate, the
intracortical circuits are low-pass and respond most strongly to slowly varying
inputs from the cortical layer-4 population.
Work was financially supported by the Research Council of Norway under the BeMatA
and eVITA programmes, and by the National Institute of Health [R01
EB00790, NS051188]. The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of the
manuscript.Author summary
Many of the salient features of individual cortical neurons appear well
understood, and several mathematical models describing their physiological
properties have been developed. The present understanding of the highly
interconnected cortical neural networks is much more limited. Lack of relevant
experimental data has in general prevented the construction and rigorous testing
of biologically realistic cortical network models. Here we present a new method
for extracting such models. In particular we estimate specific mathematical
models describing the sensory activation of thalamic and cortical neuronal
populations in the rat whisker system from joint recordings of extracellular
potentials in thalamus and cortex. The mathematical models are formulated in
terms of average firing rates of a thalamic population and a set of laminarly
organized cortical populations, the latter extracted from data from linear
(laminar) multielectrodes inserted perpendicularly through cortex. The
identified models describing the signal processing from thalamus to cortex are
found to be qualitatively very different from the models describing the
processing between cortical populations; while the thalamocortical circuit is
optimally stimulated by rapid changes in the thalamic firing rate, the
intracortical circuits respond most strongly to slowly varying inputs.
Introduction
Following pioneering work in the 1970s by, e.g., Wilson and Cowan [1] and
Amari [2] a
substantial effort has been put into the investigation of neural network models,
particularly in the form of firing-rate or neural
field models [3]. Some firing-rate network models, in particular
for the early visual system ([4], Ch.2), have been developed to account for
particular physiological data. However, for strongly interconnected cortical
networks, few mechanistic network models directly accounting for specific
neurobiological data have been identified. Instead most work has been done on
generic network models and has focused on the investigation of generic features,
such as the generation and stability of localized bumps, oscillatory patterns,
traveling waves and pulses and other coherent structures, for reviews see Ermentrout
[5] or Coombes [6].
We here (1) present a new method for identification of specific
population firing-rate network models from extracellular recordings, (2) apply the
method to extract network models for thalamocortical and intracortical signal
processing based on stimulus-evoked data from simultaneous single-electrode and
multielectrode extracellular recordings in the rat somatosensory (barrel) system,
and (3) analyze and interpret the identified firing-rate models using techniques
from dynamical systems analysis. Our study reveals large differences in the transfer
function between thalamus (VPM) and layer 4 of the barrel column, compared to that
between cortical layers, and thus sheds direct light on how whisker stimuli is
encoded in population firing-activity in the somatosensory system.
The derivation of biologically realistic, cortical neural-network models has
generally been hampered by the lack of relevant experimental data to constrain and
test the models. Single electrodes can generally only measure the firing activity of
individual neurons, not the joint activity of populations of cells typically
predicted by population firing-rate models. Kyriazi and Simons [7] and Pinto et al. [8],[9] thus
developed models for the somatosensory thalamocortical signal transformation based
on pooled data from single-unit recordings from numerous animals. By contrast,
multielectrode arrays provide a convenient and powerful technology for obtaining
simultaneous recordings from all layers of the cerebral cortex, at one or more
cortical locations [10]. The signal at each low-impedance electrode
contact represents a weighted sum of the potential generated by synaptic currents
and action potentials of neurons within a radius of a few hundred micrometers of the
contact, where the weighting factors depend on the shape and position of the
neurons, as well as the electrical properties of the conductive medium [11]–[13].
In the present paper we describe a new method for extraction of population
firing-rate models for both thalamocortical and intracortical transfer on the basis
of data from simultaneous thalamic single-electrode and cortical recordings using
linear (laminar) multielectrodes in the rat barrel system. With so called
laminar population analysis (LPA) Einevoll et al. [11]
jointly modeled the low-frequency (local field potentials; LFP) and
high-frequency (multi-unit activity; MUA) parts of such
stimulus-evoked laminar electrode data to estimate (1) the laminar organization of
cortical populations in a barrel column, (2) time-dependent population firing rates,
and (3) the LFP signatures following firing in a particular population. These
‘postfiring’ population LFP signatures were further used to
estimate the synaptic connection patterns between the various populations using both
current source density (CSD) estimation techniques and a new
LFP template-fitting technique [11].
Here we use the stimulus-evoked time-dependent firing rates for the cortical
populations estimated using LPA, in combination with single-electrode recordings of
the firing activity in the homologous barreloid in VPM, to identify population
firing-rate models. The models are formulated as nonlinear Volterra
integral equations with exponentially decaying coupling kernels allowing for a
mapping of the systems to sets of differential equations, the more common
mathematical representation of firing-rate models [5],[14].
The population responses were found to increase monotonically both with increasing
amplitude and velocity of the whisker flick [11],[15],[16]. A stimulus set varying
both the whisker-flicking amplitude and the rise time was found to provide a rich
variety of thalamic and cortical responses and thus to be well suited for
distinguishing between candidate models. The optimal model structure and
corresponding model parameters are estimated by minimizing the mean-square deviation
between the population firing rates predicted by the models and the experimentally
extracted population firing rates.
A first focus is on the estimation of mathematical models for the signal transfer
between thalamus (VPM) and the layer-4 population, the population receiving the
dominant thalamic input. For this thalamocortical transfer our experimental data
favors a model with (1) fast feedforward excitation, (2) a slower predominantly
inhibitory process mediated by a combination of recurrent (within layer 4) and
feedforward interactions (from thalamus), and (3) a mixed linear-parabolic
activation function. The identified thalamocortical circuits are seen to have a
band-pass property, and in the frequency domain the largest responses for the
layer-4 population is obtained for thalamic firing rates with frequencies around
twenty Hz.
Very different population firing-rate models are identified for the intracortical
circuits, i.e., the spread of population activity from layer 4 to supragranular
(layer 2/3) and infragranular (layer 5) layers. For the present experimental
paradigm the extracted firing rates of the various cortical laminar populations are
found to exhibit strong temporal correlations and simple feedforward models with
linear or mixed linear-parabolic activation function are found to account
excellently for the data. The functional properties of the identified
thalamocortical and intracortical network models are thus qualitatively very
different: while the thalamocortical circuit is optimally stimulated by rapid
changes in the thalamic firing rate, the intracortical circuits are low-pass and
respond strongest to slowly varying inputs.
Preliminary results from this project were presented earlier in poster format [17].
Results
Here we illustrate our approach by first showing results from one of the six
experimental data sets considered, and next show the thalamic and cortical laminar
population responses extracted from these experimental data. Further, we outline the
general form of the neural population models we explore to account for these
population firing. We then go on to test specific candidate
thalamocortical models against the experimentally observed
population responses in layer 4 using the experimental thalamic response as driving
input to the model. The identified thalamocortical models found by minimizing the
deviation between model predictions and experimental population firing rates are
further analyzed and explored using tools from dynamical system analysis. Finally,
we correspondingly examine how the experimentally measured laminar population
responses in layer 2/3 and layer 5, respectively, can be explained by
intracortical network models with the experimental layer-4
population responses as input.
Experimental data and extracted population firing rates
In Figure 1 we show
trial-averaged multi-unit activity (MUA) data for one of six
experiments, labeled ‘experiment 1’, from stimulus onset,
i.e., onset of whisker-flick, until 100 ms after. This is the time window of
data used in the further analysis. Color plots depict the laminar-electrode
recordings while the line plots below show the corresponding trial-averaged
thalamic recordings. Results for 9 of 27 stimulus conditions are shown,
corresponding to three different stimulus rise times for three different stimulus amplitudes . Neurons in the barrel cortex are sensitive to both stimulus
amplitude and stimulus velocity (and thus stimulus rise time) [11],[15],[16],
and we found that a set of stimuli varying both the amplitude and rise time
provide a rich variety of responses suitable for the present study.
10.1371/journal.pcbi.1000328.g001
Example experimental data.
Trial-averaged experimental data for experiment 1 for 9 out of 27 applied
stimulus conditions for three different amplitudes (rows) and three
different rise times (columns). Contour plots show depth profile of MUA
as recorded by the laminar electrode, normalized to its largest value
over the 22 electrode traces and 27 stimulus condition. Line plots below
show the corresponding thalamic MUA recorded by a single electrode,
normalized to the largest value over the 27 stimulus conditions.
Fig. 2 shows the estimated
thalamic and cortical population firing rates extracted from the data in Fig. 1 (procedure described in
Materials and Methods). The estimated
thalamic population firing rate is essentially just a low-pass filtered version
of the thalamic MUA. The cortical population firing rates have been estimated by
a method from the recently developed laminar population analysis (LPA)[11]
where the MUA from the full set of electrode contacts on the laminar electrode
is modeled as a sum over contributions from a set of laminar populations, cf.
Eqs. (16–17). In the present case four cortical populations are
assumed in this firing-rate extraction scheme [11], but the figure
only shows results for the three cortical populations considered in this study,
namely the layer 2/3 , layer 4 , and layer 5 populations. A deeper laminar population was also identified,
but it was left out of the present analysis since predictions related to the
postsynaptic LFP-profiles from this particular population were found to vary
between different experiments, thus questioning its proper identification [11].
10.1371/journal.pcbi.1000328.g002
Example extracted population firing rates.
Extracted population firing rates for experiment 1 for 9 out of 27
applied stimulus conditions. Green: thalamus. Black: layer 2/3. Blue:
layer 4. Red: layer 5. Note that the extracted firing rates are
normalized so that the maximum response over all stimulus conditions is
unity. For layer 2/3, layer 4 and layer 5 this occurs for the stimulus shown in the figure. For the thalamic rate it occurs
for stimulus , not shown in the figure.
Form of thalamocortical model
We first focus on the signal transfer between thalamus and the dominant input
layer of the barrel column, namely layer 4. Specifically we seek a mathematical
model predicting the layer-4 population firing rate for all 27 stimulus conditions (of which nine are depicted in
Fig. 2) given the
corresponding thalamic population firing as model input.
The stimulus-evoked dynamics of the excitatory and inhibitory neurons in layer 4
of the barrel column is complex. The excitatory neurons receive both feedforward
excitation from thalamic neurons and recurrent excitation from other layer 4
excitatory cells [9], [18]–[20].
In addition, the excitatory cells are inhibited by layer-4 interneurons [21].
These interneurons in turn receive feedforward excitation from thalamus [22]–[24] and recurrent inputs
from other layer-4 neurons [9],[21].
For mathematical convenience and conceptual simplicity we choose to formulate the
population firing-rate model as Volterra integral equations [5],[14]. In our so called
full thalamocortical model where all the abovementioned
feedforward and recurrent connections are included, we then have
Here the are temporal coupling kernels, and is a temporal convolution between the
temporal kernel and the population firing rate (Eq. 19). We here assume exponentially decaying temporal
coupling kernels (Eq. 20) which allows for a mapping of the integral equation to
a set of differential equations [5],[14].
The four terms within the large parentheses on the right hand side of Eq. (1) can
be interpreted as four contributions to the input current entering the somas of
neurons in the layer-4 population. The first and second terms correspond to
feedforward excitation and inhibition from the thalamus, respectively. The coupling kernel can be interpreted as the weight of the excitatory
contribution to the present current input to the layer-4 population due to
firing that has occurred in the thalamic population a time in the past. Likewise, the third and fourth terms correspond
to recurrent excitation and inhibition from cortical layer-4 neurons. The model structure is
illustrated in Fig. 3B. Note
that corticothalamic feedback [25] is not
explicitly modeled as this effect will only change the thalamic firing rate.
Since we do not model the thalamic firing rates and instead use the
experimentally extracted as input to the model, any effect of cortical feedback
affecting the thalamic firing rate will be included automatically.
10.1371/journal.pcbi.1000328.g003
Model structure.
(A) Illustration of form of activation function in Eq. (21). (B) Illustration of structure of
population firing-rate models with feedforward and recurrent terms in
Eqs. (1) and (10). Filled and open dots represent excitatory and
inhibitory connections, respectively.
The function in Eq. (1) is an activation function that
converts the net input current to population firing rate . This activation function is often modeled as a sigmoidal
function [4],[8], but here we found
that a simpler four-parameter threshold-type function with a linear and/or
parabolic activation above threshold provided a better fit to the experimental
data, cf. Eq. (21). The form of this activation function is illustrated in Fig. 3A. Such a threshold-type
activation function has also been seen experimentally when studying firing
characteristics of excitatory neurons in the barrel column [26].
The temporal coupling kernel is described by two parameters, a time constant and a time-delay parameter , cf. Eq. (20). To reduce the number of parameters in the model
in Eq. (1), the time delays in the recurrent temporal kernels are set to zero,
i.e., . Further, one of the weight parameters can be fixed without loss of generality. We therefore set . Nevertheless, this full thalamocortical model encompasses 13
parameters (4 for the activation function, 3 weights, 6 specifying temporal
kernels). This is a sizable number of adjustable model parameters given the
variability of the experimental data, and when comparing with our experimental
data it was observed that in particular the feedforward and recurrent inhibition
terms had strongly overlapping effects on the model predictions for the layer-4
firing rate so that their individual contributions were difficult to assess. We
thus chose to mainly investigate reduced versions of the full thalamocortical
model in Eq. (1) where either (1) only the recurrent inhibition term
(‘recurrent’ model) or (2) only the feedforward inhibition
term (‘feedforward’ model) was kept, respectively.
Recurrent thalamocortical model
We first consider the so called recurrent thalamocortical model
where the feedforward inhibitory term in the full model in Eq. (1) has been
omitted, i.e.,
This model has a special significance in that it maps (by use of the linear chain
trick [14]) to a set of differential equations that are
structurally similar to the firing-rate model suggested by Pinto et al. [8],[9], cf.
Eqs. (28–29) in Materials and
Methods.
The model was fitted to the experimentally extracted population firing rate (for all 27 stimulus conditions simultaneously) by minimizing
the mean square deviation between model and experimental results (Eq. 22) using
the optimization method described in Materials and
Methods.
Fits to recurrent model
As illustrated in Fig. 4
the recurrent model successfully accounts for the experimental observations
in all six experiments considered. In the panels each dot corresponds to an
experimentally measured layer-4 firing rate at a particular time plotted against the value predicted by the fitted model at the same time . If the model was perfect and the data noiseless, these
points should all lie on the solid line corresponding to the fitted value of
the activation function . A small spread is generally observed in Fig. 4, and this is
reflected by the low error value, i.e., in Eq. (22), of the best fits. As seen is Table 1 the errors are
all less than 6%. In Table 1 we also list the fitted parameter
values for the six experiments.
10.1371/journal.pcbi.1000328.g004
Fits of recurrent thalamocortical model.
Illustration of fits of recurrent thalamocortical model in Eq. (2) to
data from experiments 1–6. Each black dot corresponds to
the experimentally measured layer-4 firing rate at a specific time point plotted against the model value of . The red dots are corresponding experimental data
points taken from the first 5 ms after stimulus onset (for all 27
stimuli). These data points show the activity prior to any
stimulus-evoked thalamic or cortical firing and correspond to
background activity. The solid green curve corresponds to the fitted
model activation function .
10.1371/journal.pcbi.1000328.t001
Fitted model parameters for recurrent thalamocortical
model.
Exp. 1
Exp. 2
Exp. 3
(ms)
3.7
5.7
3.3
(ms)
2.5
2.0
4.5
(ms)
9.3
8.0
1.3
(ms)
13.7
14.8
30.6±0.7
4.27±0.03
2.93±0.16
1.26±0.03
4.81±0.03
3.85±0.16
1.48±0.02
0.89
0.76±0.01
0.85±0.02
0.55
0.56
0.30
1.48
1.88±0.01
0.26±0.02
−0.06
−0.05
−0.11
0.41
0.35
0.21
error
0.0522
0.0430
0.0586
Exp. 4
Exp. 5
Exp. 6
(ms)
2.8
1.9
8.2
(ms)
3.5
4.0
3.5
(ms)
14.7
15.7
3.3
(ms)
20.3
22.9
19.6
5.98±0.07
6.15±0.01
0.61
6.65±0.07
7.53±0.01
2.06
0.90
0.82
0.29
-
-
-
0.42
0.43
2.94
−0.37
−0.39
−0.13
error
0.0414
0.0371
0.0397
Resulting optimized parameters for the recurrent thalamocortical
network model (Eq. 2) incorporating feedforward excitation and
recurrent excitation and inhibition. The listed parameter values
correspond to the mean of fitted parameter values from 25
selected models giving essentially the same error (see Materials and Methods). The
standard deviations are only listed if they exceed the last
digit of the mean. Note that the value of is included to illustrate the relative
constancy of the ratio of the fitted values for and .
Fitted model parameters for recurrent model
Experiments 1–3 correspond to the stimulus paradigm with 3
different amplitudes and 9 different rise times, while experiments
4–6 correspond to 27 different amplitudes but a single fixed rise
time. As seen, for example, in Fig. 2 the variation in the rise time affects the thalamic and
cortical responses more than variation in the stimulus amplitude. Thus
experiments 1–3 exhibit a greater variation in the responses and
thus a richer data set for the models to be tested against. In fact, for
experiments 4–6 only a parabolic part of the activation function could be seen in the data, and the linear part of the
activation function in Eq. (21) was omitted from the model.
The choice of stimuli set also appears to affect the fitted parameter values
somewhat as indicated by comparing the fitted parameter values for
experiments 1, 2, 4 and 5, all carried out using the same preparation in the
same rat. As seen in Table
1 the fitted parameter values of experiment 1 and 2 are very similar,
likewise for experiment 4 and 5. The parameter values for experiments 1 and
2 also compare well with the values for experiments 4 and 5, but less so. In
the following we will focus mostly on experiments 1–3 where the
3×9 stimulus paradigm is used, and a more varied set of responses
are obtained.
For experiments 1 and 2 the fitted feedforward time constants are between 3 and 6 milliseconds and the feedforward delay between 2 and 3 milliseconds (see Table 1). The recurrent time constants
are significantly longer: between 8 and 10 milliseconds for the recurrent
excitation and between 13 and 15 milliseconds for the recurrent
inhibition . For all six experiments we observe the recurrent
inhibition to have the longest time constant. In experiments 3 and 6 the
fitted time constant of the feedforward excitation is seen to be longer than
for the recurrent excitation.
In all experiments the weight of the recurrent inhibition was found to be larger than the weight of the recurrent
excitation . However, the values of and are seen to vary significantly between the experiments.
This partially reflects that when the recurrent terms dominate the
feedforward term, large values of the recurrent weights can be compensated
by making the activation function less steep, and vice versa. However, the
ratio is found to be rather constant, typically between 0.75 and
0.9 (with experiment 6 as the only exception). Thus while excitation may
dominate at short time scales due to the shorter excitatory time constants,
inhibition dominates at longer time scales.
The positive correlation between and was confirmed by direct inspection of the correlation
matrices for the ensembles of fitted model parameters obtained during the
numerical optimization procedure (Materials
and Methods). It was also seen by inspection of the eigenvectors
of the most ‘sloppy’ direction of the
Levenberg-Marquardt Hessian (Eq. 23), i.e., the component with the smallest
eigenvalue. A ‘sloppy’ direction of means that the fitting error will change little when moving along this direction in
parameter space [27]. A positive correlation between and would imply that the fitting error will change little when
both parameters are increased (or decreased) simultaneously. Indeed, the
eigenvector of the most ‘sloppy’ eigenvalue revealed
exactly this property.
Further inspection revealed that is negatively correlated with the slope of the activation
function (parameters and in Eq. (21)). This is as expected since a decrease in the
recurrent excitatory weight can be compensated for by an increase in the
activation-function slope. Moreover, it was observed that (and ) is positively correlated to and negatively correlated to .
Phase-plane analysis of recurrent model
The identified recurrent thalamocortical models on integral form can be
mapped to a corresponding set of two differential firing-rate equations
using the ‘linear-chain trick’ [5],[14]
as described in Materials and Methods,
cf. Eqs. (25–29). This is very useful since it allows for the use
of standard techniques from dynamical systems analysis to investigate the
identified models.
With slightly redefined auxiliary variables compared to Eqs.
(25–27) in Materials and
Methods, that is,the recurrent thalamocortical model can be reformulated as
(cf. Eqs. 28–29)
This equation set resembles the model derived using
‘semirigorous’ techniques by Pinto et al. in [8],
and further explored in [9], for populations of interconnected layer-4
excitatory and inhibitory barrel neurons receiving stimulus-evoked thalamic
input.
The formulation in terms of two dynamical variables and , with only providing an external input, allows for visualization
of the circuit behavior using phase-plane analysis. In analogy with Pinto et
al. [9] we show in Fig. 5 the layer-4 model response for the
weakest and strongest stimuli for experiment 1. In the five phase plots for the
two stimuli considered the instantaneous excitatory and
inhibitory nullclines at five different times ( for weak stimulus; for strong stimulus) are shown. These nullclines are
obtained assuming a constant thalamic synaptic drive equal to the
instantaneous values or . Below the instantaneous excitatory (inhibitory) nullcline is negative, and the color code illustrates the magnitude
of .
10.1371/journal.pcbi.1000328.g005
Phase-plane analysis of recurrent thalamocortical model for
experiment 1.
Phase-plane analysis of recurrent thalamocortical model in Eqs.
(4–5) for fitted model parameters for experiment 1, cf.
Table 1.
Upper panels show the synaptic drive from thalamus and the firing rate for the weakest (; left) and strongest (; right) stimuli. The filled dots in the two upper
panels show the points in time, for and for at which the network states are shown in the two
lower panels. In each of the phase plots (five for each of the
stimuli) the instantaneous excitatory (black) and
inhibitory (blue) nullclines are shown, i.e., the nullclines
obtained with a constant thalamic synaptic drive equal to the
instantaneous value for and for . The network's activity is indicated by
the black response curve, where the head of the curve marks the
network's current state, and the tail represents prior
states.
As seen in Fig. 5 the
network model for the strong stimulus makes a significantly more extended excursion in the phase plane than the weaker stimulus . This can be readily understood by investigation of the
phase-plane plots for the times shortly after onset. For the strong stimulus
the trajectory is seen to be in a region with a large at, for example, the times and resulting in a rapid growth and large maximum value of . For these short times the fast feedforward and recurrent
excitation dominates the slower inhibition. For the weak stimulus, however,
the trajectory remains in regions with small values of (i.e., without dominant excitation). As a consequence we
observe that while the difference in maximum amplitude of the thalamic
synaptic drives between the strong and weak stimuli is seen in the upper
panels of Fig. 5 to be
less than a factor two, the difference in the maximum layer-4 firing rate is
more than a factor three. Thus the amplitude of the thalamic population
response does not explain the layer-4 response alone; the rise time of the
thalamic response is also important [9].
The qualitative conclusions from Fig. 5 is in agreement with the corresponding analysis by Pinto
et al. [9]. This is an interesting result in itself since
our model and the model of Pinto et al. [9] are derived in
very different ways. Here the parameters are extracted from a single
experiment measuring population activity directly. By contrast, in the study
by Pinto et al. the model and its parameters were chosen to qualitatively
reproduce pooled barrel population responses, i.e., single-unit poststimulus
time histograms (PSTHs), recorded from numerous animals [8],[9]. Further, the
thalamic responses used to generate their phase-plots analogous to Fig. 5 were modeled with a
simplified triangular temporal rate profile with varying rise times [9],
not extracted from experimental data like here.
Due to differences between our model (Eqs. 4–5) and the model of
Pinto et al. [9] a direct comparison between the applied
model parameters is generally not possible. However, the time constants can
be compared directly: their choice of
τEr = 5
ms and
τIr = 15
ms is qualitatively similar to what was extracted from our experiments 1 and
2, i.e.,
τEr = 8–10
ms and
τIr = 13–15
ms. Their choice of weight parameters appears to differ more from our
results: for example, they set the ratio to be between 1.7 and 2.3, i.e., stronger recurrent
excitation than recurrent inhibition. For experiments 1–3 we see
in Table 1 that the
ratio is found to be between 0.75 and 0.9, i.e., the recurrent inhibition
has a larger weight than recurrent excitation. For the
tension, i.e., the ratio of recurrent excitation over the
excitation from thalamus [8], they chose a value of about 0.9. For our
experiments this ratio showed a significant variation, varying from 1.3 to
4.3 for experiments 1–3.
Stability of background states for recurrent model
On differential form our population-firing rate model is readily available
for stability analysis. Due to the recurrent excitatory term the recurrent
thalamocortical model can in principle have unstable equilibrium points, but
the identified models must clearly exhibit stable background states when
only the stationary thalamic input (corresponding to the absence of
whisking) is present. For experiments 1–3 the background state
(cf. red dots in Fig. 4)
is found to be on the linear flank of the activation function, and the
stability of these background states can be demonstrated on the basis of the
fitted model parameters listed in Table 1. A full stability analysis for the recurrent
thalamocortical model is given in Materials
and Methods (Eqs. 30–35), and the criteria for
stability of the background state is found to be
Insertion of the fitted numerical parameter values into the inequalities Eqs.
(6–7) demonstrates that both conditions are fulfilled and that the
background state is indeed stable. Inspection of Eqs. (6–7)
further shows that instability is promoted by an increased weight of the
recurrent excitation , an increased slope of the activation function , or an increased time constant of the recurrent inhibition [28]. For example, for experiment 1 the background
state becomes unstable if is increased by a factor 1.5, by a factor 3.2, or by a factor 1.9.
Inclusion of feedforward inhibition in thalamocortical model fits
We further investigated whether the full thalamocortical model in Eq. (1), i.e.,
the recurrent model amended with the feedforward inhibition term , gives an improved fit compared to the recurrent model (Eq. 2)
alone. Numerical investigations showed that the improvement, i.e., reduction of
the error , was marginal: for experiments 1–3 the reduction in
error was in all cases less than 0.001. Thus the recurrent inhibitory term alone
seems to be sufficient to account well for the present experimental data.
We then investigated whether a pure feedforward model, i.e., a
model with feedforward excitation and inhibition only, could account for the
thalamocortical data. This model readsand in Table
2 and Fig. 6 we show
the results of the optimization.
10.1371/journal.pcbi.1000328.g006
Fits of feedforward thalamocortical model.
Illustration of fits of feedforward thalamocortical model (Eq. 8) to data
from experiments 1–3. Each dot corresponds to the
experimentally measured layer-4 firing rate at a specific time point plotted against the model value of . The red dots are corresponding experimental data
points taken from the first 5 ms after stimulus onset (for all 27
stimuli). These data points show the activity prior to any
stimulus-evoked thalamic or cortical firing and represent background
activity. The solid green curve corresponds to the fitted model
activation function .
10.1371/journal.pcbi.1000328.t002
Fitted model parameters for feedforward thalamocortical
model.
Exp. 1
Exp. 2
Exp. 3
(ms)
8.4
9.8
9.3
(ms)
2.5
3.5
5.0
(ms)
20.5±0.3
18.2±0.1
100*
(ms)
2.5
3.5
5.0
0.94
1.06
3.61
0.28±0.02
0.54±0.01
0.14
8.9±0.2
29.5±1.2
16.2±0.3
−0.15±0.01
−0.11
−0.52
−0.03
−0.00
0.02
error
0.0636
0.0691
0.0706
Exp. 4
Exp. 5
Exp. 6
(ms)
8.5
5.2
5.1
(ms)
2.5
3.0
5.0
(ms)
93.5±0.2
87.3±0.6
17.7±0.4
(ms)
2.5
1.5
5.0
1.8
2.0
1.3
0.09
0.05
0.01
3.2
1.9
8.7±0.4
−0.54
−0.76±0.02
−5.80±1.1
−0.06
−0.13
−0.07
error
0.0394
0.0351
0.0629
Resulting optimized parameters for the feedforward thalamocortical
network model (Eq. 8) incorporating feedforward excitation and
inhibition. The listed parameter values correspond to the mean of
fitted parameter values from 25 selected models giving essentially
the same error (see Materials and
Methods). The standard deviations are only listed if they
exceed the last digit of the mean. Note that for experiments
4–6 the parameter was fixed to zero in the optimization procedure.
Also, for experiment 3 the listed value 100 ms for corresponds to the maximum allowed value for this
parameter.
Comparison of fits to recurrent and feedforward thalamocortical models
For experiments 1–3, which exhibit the largest variation of responses,
the fits of the experimental data for the feedforward model are poorer than for
the corresponding fits of the recurrent model. While the errors for the recurrent model are seen in Table 1 to be 0.052, 0.043 and 0.059 for
experiments 1, 2 and 3, respectively, the corresponding errors for the purely
feedforward model are seen in Table 2 to be 0.064, 0.069 and 0.071, i.e., relative increases in
errors of 23%, 60% and 20%, respectively.
However, for experiments 4–6 where only the stimulus amplitude and not
the rise time is varied, there is no such clear preference, i.e., lower fitting
errors, for the recurrent model.
In Fig. 7 we compare the best
fits of the recurrent (Eq. 2) and feedforward (Eq. 8) thalamocortical models
with the experimentally extracted layer-4 firing rate for experiment 1 for the 9
stimulus conditions depicted in Fig. 2. Both models underestimate the peak values for the layer-4
firing rate for the shortest rise time giving the largest responses. For the stimulus giving the overall largest layer-4 response, the fitted
recurrent model both follows the rise and the ensuing fall of the response peak
better than the feedforward model, but this difference is less pronounced for
the other stimuli providing the larger responses. A detailed investigation of
the fits for individual stimulus conditions in fact shows that the fitted
recurrent model typically is closer to the experimental curves than the
feedforward model for the entire set of stimulus conditions (data not shown).
10.1371/journal.pcbi.1000328.g007
Comparison of thalamocortical model fits.
Comparison of fits of the recurrent and feedforward thalamocortical
models with experimental data set 1 for the same 9 (of 27) stimulus
conditions considered in Fig. 2. The green line corresponds to the thalamic input
firing rate , the black line to the experimentally extracted
layer-4 firing rate , while the red and blue lines correspond to the best
fits of the recurrent (Eq. 2) and feedforward (Eq. 8) models for the
layer-4 firing rate , respectively. Time zero corresponds to stimulus
onset.
The fitted activation functions for the recurrent and feedforward models for
experiments 1 to 3 are quite different (cf. Figs. 4 and 6). In particular, for experiments 1 and 2
the activation functions for recurrent models have pronounced linear flanks
which cover about half the dynamic range of . In contrast, the activation functions of the feedforward
models for these two experiments are essentially parabolic in the entire dynamic
range. In the recurrent model the ‘boosting’ of the salient
stimuli thus appears to a large extent to be provided by recurrent excitation
allowing for a partially linear activation function. In contrast, in the
feedforward model the observed amplification of strong stimuli requires a purely
parabolic activation function.
Another way to compare the candidate thalamocortical models is to investigate
their so called ‘sloppiness’ [27]. Sloppiness is
a measure of the sensitivity of the model fit to changes in model parameters:
large sloppiness means that some parameter combinations can be varied
significantly without changing the quality of the model fit. Sloppiness can be
quantified by an eigenvalue analysis of the Hessian matrix composed of the
partial 2nd derivatives of the model error function [27],[29].
Here the so called Levenberg-Marquardt Hessian matrix (Eq. 23) is used, cf. Materials
and Methods. Fig.
8 shows the eigenvalue spectra of (23) for experiments 1–3 for the three investigated
thalamocortical models: (1) the recurrent model (Eq. 2), (2) the feedforward
model (Eq. 8), and (3) the full model including both feedforward inhibition and
the recurrent connections (Eq. 1). The spectra are normalized so that the
largest eigenvalue is unity. Small eigenvalues correspond to large sloppiness,
and variations in model parameters along the direction of the corresponding
eigenvectors in parameter space will have small effects on the overall model
error [27]. The number of eigenvalues corresponds to the
dimension of . The feedforward delay parameters , which are discrete parameters in the present modeling scheme,
are absent in (cf. Materials and
Methods), and the number of eigenvalues in Fig. 8 is thus nine for the recurrent model,
seven for the feedforward model, and eleven for the full model.
10.1371/journal.pcbi.1000328.g008
Sloppiness analysis of thalamocortical model.
Illustration of parameter sensitivities, i.e.,
‘sloppiness’, of model fits for recurrent
(rec.), feedforward (feedf.) and
full (full) thalamocortical models corresponding to
Eqs. (2), (8), and (1), respectively. Eigenvalue spectra of the
Levenberg-Marquardt Hessian (Eq. 23) for experiments 1–3 are shown. The
eigenvalues are normalized to the largest eigenvalue. Delay parameters are left out of the analysis, since in our models
these parameters are set to be discrete (with a time step of 0.5
ms).
In Fig. 8 we see more
‘sloppy’ eigenvalues for the feedforward model than the
recurrent model. For example, only one of nine eigenvalues is found to be less
than 2·10−5 (measured relative to the largest
eigenvalues) for the recurrent model. In contrast, two of seven eigenvalues are
less than 2·10−5 for the feedforward model. This
larger ‘sloppiness’ is in accordance with the poorer fits
observed for the feedforward model; the functional flexibility inherent in the
feedforward model appears not to be well suited to account for the present data
with 3 different amplitudes and 9 different stimulus rise times. For the full
model we find three of eleven eigenvalues to be smaller than
10−5. Thus compared to the recurrent model, the added
feedforward inhibition with two new parameters seems only to add two
‘sloppy’ eigenvalues, in line with the observation that the
overall fit is not improved much.
To summarize, our results so far suggest that the model with recurrent inhibition
accounts slightly better for the present experimental data than the model with
feedforward inhibition, i.e., that the layer-4 interneurons providing the
inhibition of the layer-4 excitatory neurons are in turn mainly driven by
excitation from layer-4 neurons. Since the relative strength of recurrent
effects compared to feedforward effects appears to increase with increasing
thalamic stimuli [19], this might indicate that the present fits to
the experimental data might be dominated by ‘strong’ inputs.
To explore this further we investigated the properties of the thalamocortical
transfer using linear-systems measures such as transfer functions in frequency
space and impulse-response functions.
Thalamocortical transferThalamocortical frequency response
Our identified thalamocortical models are in general nonlinear due to the
nonlinear activation functions . For small deviations around a working point, however, the
above models can be linearized, so that linear-systems measures such as
transfer functions in frequency space and
impulse-response functions can be explored [30].
For the thalamocortical models the derivation of the transfer function
amounts to deriving the response to a small sinusoidal modulation in the thalamic input superimposed on a constant
background activity . As shown in Materials and
Methods the modulated response (complex notation) in layer 4 is then given by where is the linear transfer function[30]–[32]. The transfer
function for the full thalamocortical model Eq. (1) is derived in Materials and Methods, cf. Eq. (42), and
for the recurrent model in Eq. (2) it reduces to
Here and in Eq. (42) has been set to one and zero, respectively. is the derivative of the activation function at the
working point , and the Fourier-transformed coupling kernels are all given by , cf. Eq. (39).
This transfer function describes how a small modulation of the thalamic
population firing rate transfers to the layer-4 population firing rate. The
transfer function is a complex quantity where the transfer amplitude is
given by the absolute value , and the change in phase between the thalamic input and
layer-4 output is given by the complex phase angle.
Fig. 9 shows an example
of how the Fourier amplitude (lower left) and phase (lower right) of the
recurrent thalamocortical transfer-function model in Eq. (9) depends on
frequency. The parameter values correspond to the fitted values of and for experiment 1 taken from Table 1. Since the fitted activation
function is nonlinear, there is no unique derivative for the fitted model. In Fig. 9 we thus show results for two
different values of the derivative: (i) corresponding to the derivative on the linear flank, and
(ii) corresponding a larger derivative more characteristic for
the parabolic part of the activation function. In both cases the amplitude
of the model transfer function has its maximum at finite frequencies, 16 Hz
and 18 Hz, respectively. This peak stems from a resonance-like phenomenon
for frequencies where the denominator of Eq. (9) becomes small. Even larger
values of would increase the peak value further.
10.1371/journal.pcbi.1000328.g009
Thalamocortical transfer for experiment 1.
Frequency content, i.e., Fourier amplitudes, of the thalamic input (upper left) and the layer-4 firing rate (upper right) for six stimulus conditions
providing the strongest response in experiment 1 (; ; ; ; ; ). (Note that .) The corresponding amplitude (lower left) and
phase (lower right) of the experimental transfer ratio are shown as thin grey lines while the thick black
lines represent the corresponding averages for these six stimuli.
Examples of model predictions for amplitude (lower left) and phase
(lower right) of the recurrent thalamocortical transfer-function
model (Eq. 9) are shown for the fitted parameter values and for experiment 1 taken from Table 1; green lines correspond
to choosing , and dashed green lines to . The red and blue lines in the lower panels
correspond to the average of the amplitude (lower left) and phase
(lower right) of the transfer ratios found for the six stimuli
listed above when driving the fitted recurrent (red) and feedforward
(blue) models in Eqs. (2) and (8), respectively, with the
experimentally extracted thalamic input .
In Fig. 9 (lower right)
we further observe that the phase of the transfer function
is negative for the smallest frequencies. The negative transfer-function
phase implies that the maximum in the layer-4 responses will precede the
maximum of the thalamic input for a small sinusoidal input for these
frequencies.
Fig. 9 further shows the
amplitude of the Fourier components of the thalamic input (upper left), the layer-4 firing rate (upper right), as well as the amplitude (lower left) and
phase (lower right) of the experimental transfer ratio for six of the stimulus conditions providing the strongest
response in experiment 1 (). Also the average of the amplitudes and phases of the
experimental transfer ratios across these six stimulus conditions are shown.
The same characteristic resonance peak is seen in the experimental transfer
ratios as for the linear-model transfer function in Eq. (9). The theoretical
function for the largest shown value for , i.e., , is seen to be closest to the experimental curves. This is
not surprising since the depicted experimental curves correspond to six of
the stimuli giving the largest responses for which also the parabolic part
of the activation function is encountered. We see, however, that the maxima
of the experimental transfer ratios are systematically shifted to somewhat
higher frequencies compared to the predictions of the linear
transfer-function model in Eq. (9). The same small deviation is also
observed for the transfer ratio found by replacing with the Fourier transform of the best fit to the recurrent model in Eq. (2) (red curve in lower left
panel of Fig. 9). Thus
the deviation is presumably due to the non-linearities inherent in the
system under the present stimulus condition, represented by the nonlinear
activation function in the recurrent model. In Fig. 9 (lower right) we also observe a
good qualitative agreement between the predicted phase difference between
the thalamic input and layer-4 output from the linear transfer-function
model and the experimentally measured phase differences.
The experimental stimulus set with 3 different amplitudes and 9 different
stimulus rise times was found to provide the largest variation in neuronal
responses and thus provide the best test data for distinguishing the two
candidate models. As seen in Tables 1 and 2 the recurrent thalamocortical model give better fits for these
experiments, i.e., experiments 1–3, than the feedforward model.
The increases in the error measure (Eq. 22) for the feedforward model compared to the
recurrent model are between 20% and 60%. The results
for the thalamocortical transfer in experiment 1 shown in Fig. 9 indicate that the
apparent superiority of the recurrent models mainly lies in its ability to
account for the rapidly varying parts of the measured layer-4 population
response. As seen in the lower left panel of this figure, the ratio of the
frequency contents of the experimentally extracted layer-4 and thalamic
signals (solid line) is better accounted for by the recurrent model (red
line) than the feedforward model (blue line) for frequencies above 20 Hz.
For frequencies below 20 Hz the difference is less. For experiment 2, where
the increase in error from the recurrent to the feedforward model is even
larger, this tendency is even more pronounced (data not shown).
Thalamocortical impulse response
Another traditional dynamical systems measure is the impulse-response
function, i.e., the response to a very brief input signal [30]. Fig. 10 shows the impulse response of the recurrent
thalamocortical model (Eq. 2) with model parameters fitted to experiment 1,
cf. Table 1. In this
context the impulse response is the additional evoked firing in the layer-4
population when adding a sharp impulse, ‘-pulse’, to the background thalamic input. The
solid line represents the impulse response for a background activity
corresponding to the linear flank of the activation function where the strength of the -pulse is so small that the circuit remains in the linear
regime. The observed impulse response is biphasic with an excitatory phase
lasting about 15 ms followed by an inhibitory phase lasting up to 50 ms. The
dashed line shows the response to a very strong -pulse so that the nonlinear regions of the activation
function, both the parabolic and half-wave rectifying parts, are
encountered. The response to this strong excitation is seen to be
qualitatively similar to the linear impulse response. However, the
excitatory phase is somewhat sharper, and the long inhibitory phase is
modified by the half-wave rectification imposed by the requirement of
nonnegative firing rates.
10.1371/journal.pcbi.1000328.g010
Thalamocortical impulse response for experiment 1.
Impulse response for thalamocortical recurrent model (Eq. 2), i.e.,
additional response due to sharp ‘-function’-like impulse, ‘-pulse’, of extra thalamic firing
superimposed on the steady-state thalamic background activity. The
parameter values from fitting to experiment 1 are used (cf. Table 1). The
steady-state thalamic firing is set to corresponds to a steady-state
layer-4 input ‘current’ , i.e., on the linear flank of the layer-4
activation function (cf. Fig. 4). Solid line corresponds
to a linear impulse response, i.e., response to a sufficiently weak -pulse so that only the linear part of the
activation is encountered. Dashed line corresponds to a
strong -pulse of extra thalamic firing resulting in a
maximum value of , i.e., far into the non-linear region of the
activation function.
The biphasic thalamocortical impulse-response observed in Fig. 10 qualitatively
resembles the temporal shape (i.e., excitation followed by suppression) of
the ‘impulse-response’ to single whisker flicks observed
in about two thirds of the barrel-cortex neurons in a single-unit study of
Webber and Stanley [33]. The remaining third of the neurons in
this study was found to have a second excitatory phase following the
suppression [33]. It should be noted, however, that this
‘stimulus-cortical’ impulse-response is different from
the thalamocortical impulse-response extracted here, since the
‘stimulus-cortical’ response also incorporates the
processing between the whisker and the thalamus. A second excitatory phase
in the ‘stimulus-cortical’ impulse response could thus
be compatible with the observed biphasic thalamocortical impulse-response if
the thalamic response to a single whisker flick has two temporally distinct
positive phases (f.ex., due to cortical feedback onto the thalamic
neurons).
Intracortical model
Layer 4 is generally thought of as the dominant input layer for sensory
activation of cortex. The supragranular population of layer-2/3 pyramidal
neurons appears to receive a dominant input from the layer-4 population, and the
initial stimulus-evoked response in layer 2/3 appears to largely stem from layer
4 neurons which in turn are activated by the thalamic input [11],
[34]–[41]. We thus
investigated models that could account for the experimentally extracted
population firing-rate of the layer-2/3 population with the extracted layer-4
population rate as input. We initially described this activity using a combined
feedforward and recurrent model analog to Eq. (1), i.e.,
This form assumes that the interlaminar connections from layer 4 to layer 2/3 are
predominantly provided by the excitatory neurons [38]. Fitting of
this form to the experimental data revealed that the extracted population firing
rates can be explained with a strongly reduced version incorporating only the
excitatory feedforward term in Eq. (10), i.e.,
As before the weight was set to unity without loss of generality. Further, the
feedforward delay was found to be very small and consequently set to zero. Table 3 and Fig. 11 (upper row) show the
results of fitting this model to the experimentally extracted layer-2/3 firing
rates using the layer-4 firing rate as input for experiments 1–3. A first observation is
that the simple model in Eq. (11) accounts excellently for the experimental
data; the error is less than 0.03 for all experiments. The fitted time constants are all very short, less than 2 ms. The fitted activation
functions for experiments 1 and 2 have significant non-linear, i.e., parabolic,
contributions with shorter linear flanks than the corresponding activation
functions for the recurrent thalamocortical model.
10.1371/journal.pcbi.1000328.g011
Fits of intracortical model.
Illustration of fits of the intracortical model (Eq. 11) to data from
experiments 1–3. Each dot corresponds to the experimentally
measured layer-2/3 firing rate (upper row) or layer-5 firing rate (lower row) at specific time points plotted against the fitted model values of or , respectively. The red dots are the corresponding
experimental data points taken from the first 5 ms after stimulus onset
(for all 27 stimuli). These data points show the activity prior to any
stimulus-evoked thalamic or cortical firing and represent background
activity. The solid green curves correspond to the fitted model
activation functions (upper row) and (lower row), respectively.
10.1371/journal.pcbi.1000328.t003
Fitted model parameters for intracortical models.
L4→L2/3
Exp. 1
Exp. 2
Exp. 3
(ms)
1.2
1.2
1.8
0.51±0.01
0.45
0.40
0.49±0.01
0.45
0.89
−0.03
−0.03
−0.02
0.13±0.02
0.13
0.27
error ()
0.0214
0.0141
0.0279
L4→L5
Exp. 1
Exp. 2
Exp. 3
(ms)
3.0
2.1
1.4
0.96
0.86±0.01
0.73
0.24
0.10
0.27
−0.009
−0.003
−0.004
0.003
0.024±0.043
0.268±0.004
error ()
0.0195
0.0193
0.0167
Resulting optimized parameters for the intracortical network model
(Eq. 11) for experiments 1–3. Upper row: Layer-2/3 firing
rate predicted from layer-4 input. Lower row: Layer-5 firing rate
predicted from layer-4 input (where the subscripts
‘2/3’ are replaced by ‘5’ in
Eq. (11)). The listed parameter values correspond to the mean of
estimated parameters from 25 selected models giving essentially the
same error (see Materials and
Methods). The standard deviations are only listed if they
exceed the last digit of the mean.
A strong feedforward connection between the layer-4 and layer-2/3 populations is
in accordance with previous experimental studies [34]–[36], and
also with the results from the joint modeling of the present MUA data and the
corresponding LFP data using laminar population analysis (LPA)
in Einevoll et al. [11]. However, this LPA analysis also indicated a
recurrent interaction within layer 2/3, in accordance with experimental
observations by Feldmeyer et al. [42] of a substantial
connectivity between layer-2/3 pyramidal neurons. Such a recurrent connection
between excitatory neurons could amplify synchronous feedforward excitation from
layer 4 [42]. In the present modeling study we find that
no such excitatory recurrent terms are needed to account for the present
stimulus-evoked experimental data. Instead the stronger inputs are amplified in
our model by means of the boosting nonlinearity inherent in the activation
function , cf. Fig.
11. We therefore investigated to what extent recurrent connections, i.e.,
a model of the recurrent form in Eq. (10) with , improved the fit for the situation with a
linear activation function. In our test runs, it did not. For
experiment 1 with linear activation function the error was found to only be
reduced from to when adding recurrent excitation and inhibition. In
comparison, the error for the simple feedforward model with a non-linear
activation function was found to be only 0.021, cf. Table 3.
We also investigated to what the extent the extracted layer-5 population firing
rate can be accounted for by the same type of firing-rate model with the layer-4
population providing the stimulus-evoked input [43],[44].
Table 3 and Fig. 11 (lower row) show the
results of fitting a model of the type in Eq. (11) to the experimentally
extracted layer-5 firing rates using the layer-4 firing rate as input for experiments 1–3. Again we observed that
the simple feedforward model accounts excellently for the experimental data with
an error of less than 0.02 for all experiments. The fitted time constants are again short, less than 3 ms. The most striking difference
from the fits of the layer-5 firing rate is the almost linear activation
functions. Thus the layer-5 firing rate appears to be related to the layer-4
firing rate in a very simple manner for the present experimental situation.
The presently extracted intracortical models have very simple mathematical forms.
For example, for the layer-4 to layer-2/3 transfer the model on integral form
can be translated to a simple differential equation using the
‘linear-chain trick’ as described in Materials and Methods, i.e.,where the layer-2/3 firing rate is given by . The same type of differential equation is found for the
connection between layer 4 and layer 5 (with substituted by in Eq. (12)). In this case the signal transfer is essentially
described by a simple linear differential equations for since .
Intracortical transfer
We next investigated the intracortical transfer functions. The linear transfer
function from the layer-4 population to the layer-2/3 and layer-5 populations is
found from Eq. (11) to be given bywhere represents 2/3 or 5, and is found in Eq. (39). This corresponds to a low-pass filter,
i.e., no resonance behavior as for the recurrent thalamocortical model in Eq.
(9). Further, the impulse-response function is the simple (monophasic)
exponentially decaying function, contrasting the biphasic thalamocortical
impulse-response functions seen in Fig. 10.
In Fig. 12 we compare
predictions for the transfer ratio from this simple linear feedforward model
using the fitted parameters from Table 3 with the corresponding experimental result ( or 5) for six of the stimulus conditions providing the
strongest cortical responses in experiment 1 (). For the layer-5/layer-4 transfer ratio (lower row in Fig. 12) we see an excellent
agreement between experiments and linear theory. This is as expected since the
fitted layer-5 activity functions are essentially linear (cf. Fig. 11). For the transfer ratio between
layer 2/3 and layer 4 a more substantial deviation between the experimental and
linear-model transfer rations is observed, in particular a slightly growing
transfer-ratio amplitude for large frequencies. This deviation presumably
reflects the non-linearities of the layer-2/3 activity functions , because the same effect is also observed for the transfer
ratio between the best model fit and (cf. red line in upper left panel of Fig. 12).
10.1371/journal.pcbi.1000328.g012
Intracortical transfer for experiment 1.
Transfer ratios for six stimulus conditions providing the strongest
response () are shown as thin grey lines. (Note that .) Thick black lines represent the corresponding
averages for the six stimuli. Upper left: Transfer to layer 2/3,
amplitude. Upper right: Transfer to layer 2/3, phase. Lower left:
Transfer to layer 5, amplitude. Lower right: Transfer to layer 5, phase.
Examples of model predictions from the linearized intracortical
transfer-function model in Eq. (13) are shown (dashed lines) for the
parameter values corresponding to the fitted values of and for experiment 1 taken from Table 3. The work-point derivatives
were chosen to be and , respectively. The red lines correspond to the average
of the amplitude (left panels) and phases (right panels) of the transfer
ratios found for the six stimuli listed above when driving the fitted
feedforward intracortical layer-2/3 and layer-5 models of the type in
Eq. (11) with the experimentally extracted layer-4 input .
Discussion
A main objective of the present paper is the description and application of a new
method for extraction of population firing-rate models for thalamocortical and
intracortical signal transfer on the basis of trial-averaged data from simultaneous
cortical recordings with laminar multielectrodes in a rat barrel column and
single-electrode recordings in the homologous thalamic barreloid. Below we first
discuss the results from the model estimation for both the thalamocortical and the
intracortical signal transfer, followed by a discussion of the model estimation
method itself.
Thalamocortical models
For the thalamocortical signal transfer our investigations clearly identifies a
model with (1) fast feedforward excitation, (2) a slower predominantly
inhibitory process mediated by recurrent (within layer 4) and/or feedforward
interactions (from thalamus), and (3) a mixed linear-parabolic activation
function. The relative importance of the feedforward compared to the recurrent
connections is more difficult to determine and has been a subject of significant
interest [7]–[9],[16],[45].
Here we have compared two alternative models: (i) a purely
feedforward thalamocortical model with feedforward excitation
and inhibition and (ii) a recurrent model with feedforward
excitation, recurrent excitation and recurrent inhibition (but no feedforward
inhibition). In reality the inhibitory layer-4 neurons receive excitatory inputs
both from thalamic neurons and other layer-4 neurons, and the inhibition felt by
the excitatory layer-4 neurons will thus be both
‘feedforward’ and ‘recurrent’; the
different models thus explore what part appears to dominate in the present
experimental situation.
Our results suggest that the model with recurrent inhibition better accounts for
the present experimental data than the model with feedforward inhibition, i.e.,
that the layer-4 interneurons providing the inhibition of the layer-4 excitatory
neurons are in turn mainly driven by excitation from layer-4 neurons. Since the
relative strength of recurrent effects compared to feedforward effects appears
to increase with increasing thalamic stimuli [19], this might
indicate that the present fits to the experimental data is dominated by
‘strong’ inputs. Alternatively, it may be that a strong
feedforward inhibition is also present, but that it is strongly overlapping in
time with the feedforward excitation. Then the present modeling scheme and
experimental data are unable to separate them and instead lump the effect of
this inhibition together with the feedforward excitation.
An interesting result from the present investigation is that we find a similar
thalamocortical population model as Pinto et al. [8],[9]: the
recurrent model on integral form in Eq. (2) can be mapped
to a set of two differential equations (Eqs. 4–5) structurally similar
to the model suggested and investigated by Pinto and coworkers. The
correspondence between the behavior of these models is demonstrated by the
phase-plane plots in Fig. 5,
which exhibits the same qualitative features as the analogous phase-plane plots
in Ref. [9]. This close correspondence is striking since the
methods used to derive the models are very different. Pinto et al. chose the
model to account for a large collection of single-unit PSTHs recorded with sharp
electrodes from numerous animals. By contrast, our model parameters were
extracted based on simultaneous thalamic and cortical recordings from individual
animals. As multielectrode arrays allow for convenient simultaneous recordings
from all layers of the cerebral cortex, even at more than one cortical location
[10], the proposed modeling approach scales more
readily to complex networks involving multiple, spatially separated, neuronal
populations.
Intracortical model
The presently identified intracortical model for the connection between the
layer-4 and layer-2/3 populations has a very simple mathematical form including
only a single feedforward term (Eqs. 11–12), but was nevertheless
found to account excellently for the experimental data, cf. Fig. 11. Such a strong feedforward connection
between these populations is in accordance with previous experimental studies
[34]–[36]. An experimental
study by Feldmeyer et al. [42] found substantial recurrent connectivity
between layer-2/3 pyramidal neurons, but no such excitatory recurrent terms were
needed to account for the present stimulus-evoked experimental data. Compared to
the thalamocortical transfer, the intracortical transfer displayed limited
variation for the present set of stimuli under the present experimental
conditions. The absence of identified recurrent terms in our intracortical
models may thus be due to lack of sufficient variance in the experimental data
to allow for identification of such a model term. Instead recurrent
intracolumnar interactions in layer-2/3 may be involved in shaping the
non-linear activation function and thus modifying the dynamics in a way which cannot be
accounted for by the present recurrent model with a linear activation function.
As seen in Fig. 11 and Table 3 a simple feedforward
model can, based on the layer-4 population firing rate, also account excellently
for the observed layer-5 population firing rate. As for the layer-4 to layer-2/3
pathway, the fitted time constants are quite short, 3 ms or less. Interestingly,
the fitted activation function is found to be almost linear (, cf. (12)), so that the dynamics of the layer-5 population
firing rate can be described simply by . Monosynaptic connections between pairs of spiny stellate
cells in layer 4 onto the basal dendrites of layer-5 pyramidal cells compatible
with a such a fast feedforward connection have been observed experimentally
[43],[44]. Moreover,
results from the LPA analysis of the present data in Einevoll et al. [11]
were compatible with such an excitatory synaptic connection from layer 4 onto
the basal layer-5 neuron dendrites. However, the presently extracted
intracortical firing-rate model may simply reflect that extracted firing rates
for the cortical populations are strongly overlapping in time. An alternative
explanation for the observed tight temporal correlation between the layer-4 and
layer-5 firing rates could be correlations between the input from thalamic
neurons in VPM onto layer 4 with VPM inputs to layer-5B neurons [46]
and/or ‘paralemniscal’ input via POm to layer 5A neurons
[39],[47]. Thus while the
thalamocortical transfer seems to be well probed by the present stimulus
paradigm, i.e., the set of stimuli provides a varied set of paired thalamic and
layer-4 responses, this is less so for the intracortical transfer. One should
thus search for other experimental paradigms to probe the intracortical
population-rate dynamics further.
As a methods test we also investigated to what extent our feedforward model could
account for the ‘backwards’ connections, i.e., whether the
layer 4 firing could mathematically be accounted for by the model in Eq. (11)
with the extracted layer-2/3 or layer-5 population firing rates as input
respectively. These connections are expected to be weak [37],[41],
and indeed, since the layer-4 firing initiates somewhat earlier than the rest of
the laminar populations, the best fits were found for unrealistically small time
constants (τEf<0.1 ms). The model
solutions naturally failed to predict the onset of the layer-4 firing correctly,
indicating their inadequacy.
Coding properties of thalamocortical and intracortical circuits
A conclusion from our study is that the signal processing in the thalamocortical
and intracortical circuits is very different. The thalamocortical circuit favors
rapid whisker movements in that sharply rising thalamic population firing rates
evokes large cortical responses. For such stimuli the fast excitation in the
circuit dominates the slower, but eventually stronger, inhibition at short time
scales thus ensuring a large population firing-rate in the layer-4 population
for a certain time window. For slower whisker movements the inhibition overtakes
the excitation before a strong cortical response has emerged. This essential
feature is demonstrated by the phase-plane plots in Fig. 5. In frequency space the effect
translates into a resonance-like amplification of the transfer function at
intermediate frequencies (∼15–20 Hz), cf. Fig. 9, while the impulse response function
is biphasic, cf. Fig. 10. In
terms of coding the thalamocortical transfer can be said to map time derivatives
of the thalamic population firing rate onto amplitudes in the population firing
of cortical layer 4 [9],[16]. Thus the
thalamocortical circuit is sensitive to angular whisker velocity, explaining why
a rich variety of cortical responses are observed for the stimulus set with
three different amplitudes and nine different rise times.
The intracortical circuits appear very different. Here the extracted models
perform low-pass filtering, but the time constants are so short, typically
1–3 ms, that the cutoff frequency is large: for firing-rate
frequencies up to 50 Hz only modest reductions in the transfer ratio is
observed, cf. Fig. 12. In
contrast to the typical biphasic form seen for the impulse response for the
thalamocortical model in Fig.
10, the impulse response of the intracortical models is a rapid
exponentially decaying (monophasic) function. Thus, while the thalamus to
layer-4 circuit appears to selectively respond to synchronous inputs with large
time derivatives, the layer-4 to layer-2/3 circuit appears to perform a simple
amplitude mapping [16].
Current limitations and potential future extensions
A core assumption in our modeling scheme is that the MUA signal is proportional
to the population firing rate [48],[49].
This assumption was recently investigated by Pettersen et al. [12]
in a forward-modeling study of extracellular potentials from a population of
layer-5 pyramidal cells from cat visual cortex receiving synchronous synaptic
activation resembling the present stimulus-evoked situation. For
trial-averaged data a roughly linear relationship was
observed for a large range of small and intermediate population firing rates.
For large population firing rates the true population firing rate was found to
grow superlinearly with the MUA due to increasing cancelation of extracellular
potentials for high firing rates. It is, however, unclear whether or to what
extent such a superlinear relationship can be expected here.
A number of factors in our study could affect the magnitude and frequency content
of the recorded electrophysiological response in comparison with awake behaving
animals in a natural environment. Among them are anesthesia conditions and the
use of an artificial single-whisker stimulus in the absence of active whisking.
The present study used alpha-chloralose, an anesthetic agent used widely in
neurophysiological and hemodynamic studies due to its lesser effects on
cardiovascular, respiratory and reflex functions [50]–[52].
Qualitatively similar laminar profiles of evoked MUA response was observed with
ketamine [11], but a systematic study is needed to
investigate the effects of anesthesia on the identified network models.
Presently we have assumed the measured MUA to correlate with the excitatory
population. Due to their lower neuron numbers the inhibitory populations are
expected to contribute less, although differences in morphology and firing rates
between excitatory and inhibitory neurons will also affect their relative
contributions [13]. More forward-modeling studies along the
lines of Pettersen et al. [12] are needed to investigate this relationship
between the measured MUA and the underlying neural activity in intermixed
excitatory and inhibitory populations.
In the current models only the firing rates of the excitatory laminar populations
are modeled explicitly, while the inhibitory neurons are merely considered to
modify the dynamics of, and interaction between, these excitatory populations.
An alternative to the single population Volterra firing-rate model for the
thalamocortical transformation in Eq. (1) could be the following generalization
including an explicit dynamical equation also for the inhibitory firing rate, i.e.,
This model explicitly includes thalamic input to both the excitatory and
inhibitory populations as well as all possible recurrent connections [9]. The
two integral equations in Eqs. (14–15) can be mapped to a set of four
differential equations using the ‘linear-chain trick’ [14].
However, as shown in Materials and Methods,
the model in Eqs. (14–15) can be reduced to our model in Eq. (1) under
the assumptions that (1) the inhibitory firing-rate function is linear, i.e., , (2) the self inhibition is zero, [53],
and (3) by making the identifications , and , cf. Eqs. (43–47).
The analysis of model parameter identifiability revealed that certain linear
combinations of parameters (notably recurrent and feedforward excitation and
inhibition strengths, time constants, and activation function parameters), do
not have a measurable effect on the measured signals, and thus cannot be
confidently estimated. This problem can be partly addressed by using
intracellular recordings to measure some of these parameters, e.g., synaptic
time constants and firing rates as function of somatic input currents, directly
in the relevant cell types. Model identification could also be aided by
theoretical investigations of the basic structure of population firing-rate
models [54],[55].
Furthermore, intercolumnar signal propagation can be modeled through connecting
single-column firing-rate models of the present type based on known connectivity
between neighboring columns or commissural projections [56]–[62].
Such multicolumn models may provide insights into the mechanisms of more complex
cortico-cortical interactions such as surround [63],[64] and
transcallosal inhibition in SI [65]–[67].
Several groups have attempted to derive information about functional connectivity
between cortical areas based on non-invasive imaging data [68],[69]. A
further development of this approach, aimed at providing a general theory of
information processing in hierarchies of cortical areas, has recently been
proposed by Friston [70],[71]. The
interpretation of the results from such high-level modeling approaches has been
limited by the lack of a biophysical measurement theory relating the imaging
signals to the underlying neuronal population activity and the connections
between neuronal populations. The modeling approach described here could provide
the necessary biophysical grounding for such theories, as the noninvasive
electroencephalography (EEG) and magnetoencephalography (MEG) signals can be
directly predicted based on the laminar distribution of current sources and
sinks, along with information about the geometry of the cortical surface of the
individual subject [72],[73]. Thus, by embedding
the proposed ‘mesoscopic’ modeling approach within a
macroscopic modeling framework, such as the Hierarchical Dynamical Model (HDM)
proposed by Friston [71], it may ultimately be possible to bridge the
gap from the cellular to the systems-level of description, and from invasive to
non-invasive recordings.
Materials and MethodsExperimental procedure
All experimental procedures were approved by the Massachusetts General Hospital
Subcommittee on Research Animal Care. Male Sprague-Dawley rats
(250–350 g, Taconic) were used in the experiments. Glycopyrrolate (0.5
mg/kg, i.m.) was administered 10 minutes before the initiation of anesthesia.
Rats were anesthetized with 1.5% halothane in oxygen for surgery.
Following surgery (see below), halothane was discontinued, and anesthesia was
maintained with 50 mg/kg intravenous bolus of alpha-chloralose followed by
continuous intravenous infusion at 40 mg/kg/h. During surgery a tracheotomy was
performed, cannulas were inserted in the femoral artery and vein. All incisions
were infiltrated with 2% lidocaine. Following tracheotomy, rats were
mechanically ventilated with 30% O2 in air. Ventilation
parameters were adjusted to maintain the following blood gas readings:
PaCO2 between 35 and 45 mm Hg, PaO2 between 140 and
180 mm Hg, and pH between 7.35 and 7.45. Heart rate, blood pressure, and body
temperature were monitored continuously. Body temperature was maintained at
37.0±0.5°C with a homeothermic blanket (Harvard Apparatus,
Holliston, MA, USA). The animal was fixed in a stereotaxic frame. An area of
skull overlying the primary somatosensory cortex was exposed and then thinned
with a dental burr. The thinned skull was removed and the dura matter dissected
to expose the cortical surface. A barrier of dental acrylic was built around the
border of the exposure and filled with saline.
Mapping with a single metal microelectrode (FHC, 2–5 MΩ) was
done to determine the positioning of the linear (laminar) multielectrode array
in barrel cortex. The optimal position was identified by listening to an audio
monitor while stimulating different whiskers. Following the mapping procedure,
the electrode was withdrawn, and the laminar electrode was slowly introduced at
the same location perpendicular to the cortical surface. Contact no. 1 was
positioned at the cortical surface using visual control with saline covering the
exposed cortex around the laminar electrode. The linear multielectrode had 23
contacts with diameter 0.04 mm spaced at 0.1 mm [49], and data from
contacts 2 to 23 was used in the further analysis.
The thalamic recordings were performed using single metal microelectrodes (FHC,
5–7 MΩ). The ventral posteriomedial thalamic nucleus (VPM)
was targeted using stereotactic coordinates (VPM: AP −3.6 to
−3.0, ML 2.0 to 3.5, DV 5.0 to 7.0). VPM (rather than POm) was chosen
because it provides the dominant input to cortical layer 4 [74].
Single whiskers were deflected upward by a wire loop coupled to a
computer-controlled piezoelectric stimulator. The stimulus sequence was
optimized using the approach described by Dale [75]. This method
optimizes efficiency for Finite Impulse Response (FIR) estimation, and, thus,
makes no a priori assumption of the response shape. Stimuli were presented at a
fixed rate of one per second, with 25% of stimuli being
‘null events’ (zero amplitude whisker flicks). The method
was used to optimize the order of stimulus conditions so as to optimize
estimation efficiency of the event-related neuronal responses. We used two
stimulus conditions: In the first stimulus condition, with altogether 27
different stimuli, three stimulus amplitudes (vertical displacements 0.40 mm (), 0.80 mm (), and 1.2 mm ()) each with nine stimulus rise times (20 ms (), 30 ms (),…, 100 ms ()) were applied. The stimulus angular velocity varied between
76 deg/s () and 1090 deg/s (). In the second stimulus condition a fixed stimulus rise time
(time from onset to maximum displacement) of 30 ms was used, with 27 different,
linearly incrementing, vertical displacements up to 1.2 mm (amplitude 27).
Preprocessing of experimental data
The recorded laminar-electrode potential was amplified and analogically filtered
online into two signals: a low-frequency part and a high-frequency part [49].
Only the high-frequency part (150–5000 Hz, sampled at 20 kHz with 12
bits) was used in the present analysis. This signal was further filtered
digitally between 750 Hz and 5000 Hz using a zero phase-shift second-order
Butterworth filter, and then rectified along the time axis to
provide the multi-unit activity (MUA). This non-negative high-frequency
‘envelope’ reflects firing of action potentials [12],[48],[49].
The time-resolution of this MUA signal was then decimated by a factor 10,
reducing the time resolution of the data from 0.05 ms to 0.5 ms [11].
Sample traces of the laminar-electrode MUA (prior to rectification) for a single
trial can be seen in Figure
1B in Einevoll et al. [11]. The process of (1) band-pass filtering
(750–5000 Hz), (2) rectification, and (3) decimation was also applied
to the thalamic single-electrode recordings to provide the thalamic MUA signal.
The presently used trial-averaged data were obtained by
averaging over all 40 trials for each stimulus type.
Six experiments recorded from a total of three rats were considered. In
experiments 1–3 the stimulus condition with 3 amplitudes and 9
rise-times was used, while the 27-amplitude stimulus condition was used in
experiments 4–6. Experiments 1, 2, 4 and 5 were from a single rat,
while experiments 3 and 6 were from two other rats. About ten (out of 6480)
trials were removed due to artifacts in the laminar-electrode MUA. In
experiments 3 and 6, the laminar-electrode MUA signal from contact no. 9 was
erratic; for this channel the arithmetic mean of the MUA signals from the
neighboring contacts (contacts no. 8 and 10) was therefore used instead. The
trial-averaged MUA data used for the estimation of cortical population firing
rates is denoted . and refer to the electrode position and time, respectively, and
the indices and run over electrode contacts and time points.
Estimation of thalamic population firing rate
The MUA recorded by the single thalamic electrode was used as a measure of the
population firing rate of thalamic neurons in a barreloid in VPM projecting onto
the homologous barrel. An estimate of the trial-averaged thalamic firing rate was found by an additional low-pass filtering of the MUA using
a zero phase-shift, third order Butterworth filter with a cutoff frequency of
200 Hz. With this filter a -function is transformed to a peak with a half-width of about 2
ms. The value of the resulting rate signal was then shifted and normalized in
each experimental data set separately so that the minimum and maximum estimated
(normalized) thalamic firing rate over all times and stimulus conditions was
zero and unity, respectively.
Estimation of cortical population firing rates
The estimation of population firing rates of laminar cortical populations from
laminar-electrode MUA is less straightforward. However, Einevoll et al. [11]
recently developed a new method, laminar population analysis
(LPA), for analysis of laminar-electrode data. Using this method,
laminarly organized neuronal populations can be identified and the time
dependence of the population firing rates estimated. The laminar-electrode MUA
data are then modeled as a sum over spatiotemporally separable
contributions from several neuronal populations, i.e.,where is the number of populations, is the MUA spatial profile related to firing of action
potentials in neuronal population , and relates directly to the corresponding time course of firing
activity in this population. With additional physiological constraints imposed
on the general shape of , both and can, as described in Einevoll et al. [11], be determined in
an optimization scheme minimizing the square deviation between and the experimental data . In Einevoll et al. [11] four distinct
cortical populations were identified and putatively interpreted as layer 2/3,
layer 4, layer 5, and layer 6 (and/or deep layer 5) populations. Experiments 1,
3, 5, and 6 correspond to data also analyzed there, and fitted results for from this study were applied directly. For illustration of the
correspondingly estimated spatial profiles , see Fig.
12 in Ref. [11]. For experiments 2 and 4 new LPA analyses
were carried out to extract .
Pettersen et al. [12] recently investigated the relationship
between action-potential firing in a neuronal population and the MUA recorded by
adjacent laminar electrodes. With a forward modeling scheme they calculated the
extracellular potential generated by a columnar population of layer-5 pyramidal
cells following synaptic activation resembling the stimulus-evoked situation
investigated here. They found that a filtered version of the raw MUA in general
gave a good measure of the population firing rate, known exactly in their model
situation, for the trial-averaged case. Their study thus support the use of Eq.
(16). Further, in their forward-model study a Gaussian filter with a standard
deviation of 1 ms was found to be suitable. Presently, the signal was low-pass
filtered at 200 Hz with a zero phase-shift, third order Butterworth filter,
giving a similar temporal filter width. Fig. 13 demonstrates the effect of this
low-pass filtering on example raw MUA thalamic and layer-4 population signals
both for a weak () and a strong () stimulus condition.
10.1371/journal.pcbi.1000328.g013
Effects of filtering on population firing-rate estimates.
Example trial-averaged firing-rate traces, both unfiltered and filtered,
from experiment 1 for the thalamic and layer-4 populations for a weak () and a strong stimulus (). The unfiltered layer-4 traces correspond to , cf. Eq. (16), while the layer-4 filtered traces
correspond to the firing-rate estimate described in Eq. (17). The unfiltered thalamic traces
represent the raw single-electrode MUA data. The relationship to the
thalamic firing-rate estimate is described by Eq. (17).
The resulting filtered estimateswere then shifted and the constant adjusted so that in each experimental data set the minimum
firing rate was zero and the maximum firing rate unity for each identified
cortical population. Here the superscript ‘’ denotes that the firing-rate estimates stem from
experiment, and the superscript ‘’ denotes that the signal has been low-pass filtered.
We assume that is a measure of the firing rate of laminar populations of
excitatory neurons. For example, is interpreted as the population firing rate of the excitatory
neurons in layer 4 of the barrel column. There will certainly also be
contributions to the MUA from the inhibitory neurons, but since more than
80% or neurons in barrel cortex are excitatory [39] and the excitatory
and inhibitory neurons in layer 4 have been observed to have firing
probabilities with similar size and temporal profile when stimulated by the
principal whisker [18], we assume the recorded signal to correlate
well with the firing-rate of the excitatory population.
Form of population firing-rate model on integral form
The population firing-rate models are formulated as nonlinear Volterra integral
equations [5],[14],where is a temporal coupling kernel, and is the temporal convolution given by
The quantity can be interpreted as net input current entering the somas of
neurons belonging to population . This current is assumed to result from previous firing in all
presynaptic populations (possibly including itself), with and describing the excitatory and inhibitory effects,
respectively, of firing at different times in the past. Presently, the temporal
coupling kernels are modeled as (normalized) delayed decaying exponentials, i.e.,where is the Heaviside unit step function. This kernel is specified
by two parameters, the time constant and the time delay .
In our model the net input current is converted instantaneously into a population firing rate by means of a nonlinear function . Here the following four-parameter class of activation
functions is used:
As illustrated in Fig. 3A
this activation function is zero below the threshold , grows linearly between and , and quadratically above . The activation function is continuous, and the positive
coefficients and determine the steepness of the linear and quadratic term,
respectively. Note that for (or ) the linear part of the activation vanishes. Likewise, for (or ) the activation above threshold is linear.
Numerical optimization procedure
The model parameters were estimated by fitting model estimates of the population
firing rates to the corresponding experimentally extracted rates for all stimulus conditions simultaneously. The squared
deviation between model and experimental firing rates was used as error measure, i.e.,
Here the sums over and go over all time steps, i.e., 201 for each stimulus multiplied
by the 27 different stimuli, and denotes the temporal mean of .
The numerical calculations were done using MATLAB. In the numerical evaluation of
the thalamocortical model (Eq. 1) the experimentally extracted thalamic firing
rate was inserted directly. The convolutions in the feedforward
terms on the right hand side of Eq. (1) could thus be performed once and then
stored for use in the later numerical procedure. For the recurrent term this is
not possible since is evaluated iteratively for each time step. The firing rate at time is given by the right hand side of Eq. (1) with the recurrent
convolution terms evaluated at time where
Δt = 0.5 ms is the
time step. As a result, the recurrent synaptic connections can be considered to
have a fixed time delay of 0.5 ms. The convolutions with the exponential
coupling kernels were approximated using MATLAB's FILTER command. No
explicit delays were considered for the recurrent terms, but for the feedforward
terms delays corresponding to multiples of the time step (0.5 ms) were allowed
for.
A three-step optimization procedure was used. In the first step the full model
was considered and a preliminary optimization was performed doing a stochastic
search starting from several sets of random initial parameter values where represent individual model parameters. We then computed linear
correlations between the parameters for an ensemble of fitted models providing best fits, i.e., lowest values of the error in Eq. (22), from this initial search. These correlations were
then used to temporarily reduce the number of independent parameters to be used
in a second stage of the optimization. For this second stage initial parameter
values were chosen randomly from a distribution around the mean parameter values
obtained from the ensemble of first-stage fits. After the second round of
optimization in the reduced parameter space, all model parameters were again
allowed to vary independently in the full parameter space. In this third part of
the optimization procedure the initial values were taken to be the parameters
resulting from the second-stage run. MATLAB's FMINSEARCH command, based
on the Nelder-Mead algorithm, was used at the second and third stages of the
optimization procedure. After the establishment of the ensemble of first-stage
fits, a set of 25 independent second (and subsequent third) stage optimizations
was run. The optimization procedure terminated if all 25 optimization jobs
obtained the same minimum error value within 0.0001. If not, the procedure
starting from the second stage, was repeated.
Analysis of model ‘sloppiness’
In the optimization procedure we take advantage of linear correlations in the
ensembles of fitted model parameters. Principal component analysis (PCA) of the
covariance matrices , where and denotes the ensemble average, showed a substantial
interdependency between these fitted parameters. More than 99% of the
variability of the fitted parameters was typically accounted for by the first
two components. As a consequence, large variations in the sets of optimal
parameters may leave the overall quality of the fit, i.e., the error measure,
almost unaffected as long as they are restricted to the subspace spanned by the
dominant principal components. This property is called
sloppiness, and can be quantified by an eigenvalue analysis of
the Hessian matrix composed of the partial 2nd derivatives of the model error
function evaluated for the estimated optimal (minimum error) parameter
combination [27],[29]. To evaluate the
sloppiness of our various candidate models the so called Levenberg-Marquardt
approximation of the Hessian, defined via [27],[29]was used, where the derivatives were evaluated for the overall
best parameter set . Here is the model firing rate at time , and the sum goes over all data points, in our case
201×27 = 5427. Note that the delay
parameters (and for the models with feedforward inhibition) were left out of
the analysis, since in our models the smallest possible step in this parameter
was 0.5 ms. The depicted eigenvalues of shown in Fig.
8 were normalized by the maximum eigenvalue.
The spectrum of these eigenvalues can be related to the shape of an ellipsoidal
constant-error surface in a high-dimensional parameter space. The length of each
principal axis of this constant-error surface is inversely proportional to the
square root of the corresponding eigenvalue [27]. Directions
with large eigenvalues are thus ‘stiff’ in the sense that
the error increases rapidly in this direction. In contrast, in the
‘sloppy’ directions, corresponding to small eigenvalues, the
error value changes little.
Thalamocortical model as differential equationsDerivation of differential equations
The full thalamocortical model on integral form in Eq. (1) can be mapped to a
set of two differential equations by means of the ‘linear chain
trick’ [5],[14]. This model can
be written aswhere the auxiliary variables and are given by
Since the temporal coupling kernels all are given as exponentially decaying
functions, differentiation of the expression for in Eq. (25) and in Eq. (26) with respect to time yields [14],[76]
Here we have used that the delay constants and in the recurrent coupling kernels and are set to zero.
Stability analysis of thalamocortical model
On differential form the stability of the thalamocortical model can be
directly assessed. For the situation with a stationary input , the differential equation system in Eqs.
(28–29) is given by
The Jacobian of this autonomous equation system is given bywhere . To ensure stability of the equilibrium point for this
two-dimensional system, a negative trace tr() and a positive determinant det() of the Jacobian is required (see, e.g., Wyller et al.
[28]). For our activation function in Eq. (21) we have
For the situation where the equilibrium point corresponds to on the linear flank of the activation function, we thus
have at the equilibrium value . We then findwhere we have used that and . The requirements of a negative-trace and a
positive-determinant for stability of the equilibrium point thus translate
to the inequalities in Eqs. (6–7) in the main text.
These expressions can be directly generalized to the case where the
background equilibrium state is on the parabolic part of the activation
function by replacing by . Without recurrent excitation, i.e., , stability is always assured.
Thalamocortical transfer function
Our thalamocortical models are in general nonlinear due to the nonlinear
activation function described by Eq. (21). However, a linear transfer function can
be found by linearization around a working point, i.e., by deriving the response
to a small sinusoidal modulation in the thalamic input population firing rate superimposed on a
constant background firing rate . Following a standard perturbative approach we make the ansatz
With these expressions the convolutions in Eq. (1) can be rewritten aswhere we have used that (i) is normalized, (ii) by construction so that the lower boundary in the integral
above can be set to , and (iii) introduced the Fourier transform . For our exponentially decaying coupling kernels in Eq. (20)
we further find
We now assume the perturbations in the input to be small and make the Taylor expansionso that we can identify . By using the general mathematical result in Eq. (38) in the
model expression Eq. (1) we thus find
Algebraic rearrangement of this expression gives where is given by
Reduction of two-population thalamocortical model
An alternative to the one-population Volterra model in Eq. (1) could be a
generalization to a two-population Volterra model where both excitatory and
inhibitory population firing rates are modeled explicitly. For the
thalamocortical transformation this translates into including both an excitatory
and an inhibitory layer-4 population, cf. Eqs. (14–15). With certain
assumptions, however, this two-population model reduces to the one-population
model in Eq. (1): with a linear inhibitory activation function,
i.e., , and no self-inhibition, i.e., [53],
we find from Eqs. (14–15) that
Then a term-by-term comparison shows that this expression has identical form as
Eq. (1) provided the identificationsare made.
We thank John Wyller and Harvey Swadlow for useful discussions, Klas Pettersen for
assistance with the figures, and Tom Tetzlaff and Ad Aertsen for critical reading of
an early version of the manuscript.
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