Prediction.

Given the estimated state , the observation and inputs and , we predict the state variables and the covariance matrix of prediction error of the system at time .where and

Updating.

We use the new observation at time to update the state of the system.where

Animals and visual discrimination training.

Three female sheep were used (Ovis aries, one Clun Forest and two Dorsets). All experiments were performed in strict accordance with the UK 1986 Animals Scientific Procedures Act. During the experiments the animals were housed inside in individual pens. They were trained initially over several months to perform operant-based face (sheep) or non-face (objects) discrimination tasks with the animals making a choice between two simultaneously presented pictures, one of which was associated with a food reward. During stimulus presentations, animals stood in a holding trolley and indicated their choice of picture by pressing one of two touch panels located in the front of the trolley with their nose. The food reward was delivered automatically to a hopper between the two panels. The life-sized pictures were back projected onto a screen in front of the animal using a computer data projector. A white fixation spot on a black background was presented constantly in between trials to maintain attention and experimenters waited until the animals viewed this spot before triggering presentation of the image pairs. The stimulus images remained in view until the animal made an operant response (generally around ). In each case, successful learning of a face or object pair required that a performance criterion of correct choices over 40 trials (i.e. 40 pairs) was achieved consistently. By the end of training, animals were normally able to reach the correct criterion after 40–80 trials and maintain this performance. For the current analysis extensive recordings taken during and after learning of novel face pairs were used (2 pairs for Sheep A; 2 pairs for Sheep B – only one of which was successfully learned – and 1 pair for Sheep C). In all cases recordings were made over 20–80 trials during learning and then during 46–170 trials after the correct criterion was reached. Post learning trials ranged from within 5–10 minutes of the end of a learning trial session to 2 months after learning. For the face pairs, Sheep A and B were discriminating between the faces of different unfamiliar sheep faces (face identity discrimination) whereas for Sheep C, discrimination was between calm and stressed face expressions in the same animal (face emotion discrimination). With this latter animal, the calm face was the rewarded stimulus.

Electrophysiological recordings and analysis.

Following initial behavioral training sheep were implanted under general anesthesia (fluothane) and full aseptic conditions with unilateral (Sheep A-right IT) or bilateral (Sheep B and C) planar 64-electrode arrays (epoxylite coated, etched, tungsten wires with spacing - total array area around tip diameter , electrode impedence ) aimed at the IT. Holes ( diameter) were trephined in the skull and the dura beneath cut and reflected. The electrode bundles were introduced to a depth of from the brain surface using a stereotaxic micromanipulator and fixed in place with dental acrylic and stainless-steel screws attached to the skull. Two of these screws acted as reference electrodes, one for each array. Electrode depths and placements were calculated with reference to X-rays, as previously described. Electrodes were connected to 34 pin female plugs (2 per array) which were cemented in place on top of the skull (using dental acrylic). Starting 3 weeks later, the electrodes were connected via male plugs and ribbon cables to a 128 channel electrophysiological recording system (Cerebus 128 Data Acquisition System - Cyberkinetics Neurotechnology Systems, USA) and recordings made during performance of the different face and non-face pair operant discrimination tasks. This system allowed simultaneous recordings of both neuronal spike and local event-related (LFP) activity from each electrode. Typically, individual recording sessions lasted around 30 min and for 80–200 individual trials. There was at least a week between individual recording sessions in each animal. The LFPs were sampled at 2kHz and digitized for storage from around 3 seconds prior to the stimulus onset to around 3 seconds after the stimulus onset (stimulus durations were generally ).

For data analysis of the stored signals LFP data contaminated with noise such as from animal chewing food were excluded as were LFPs with unexpectedly high power. For LFPs, offline filtering was applied in the range of and trend was removed before spectral analysis. Any trial having more than 5 points outside the mean standard deviation range were discarded before analysis. At the end of the experiments, animals were euthanized with an intravenous injection of sodium pentobarbitone and the brains removed for subsequent histological confirmation of X-rays that array placements were within the IT cortex region.

Toy Model 1.

The first toy model we used comes from a traditional VAR model which has been extensively applied in tests of Granger causality [31]. We modified the model by adding two deterministic inputs and . was assumed to be a constant stimulation, i.e. while was assumed to a harmonic oscillator and had the form of a sinusoidal function since biological rhythms are a common phenomenon. Observation variables were also included in the toy model and assumed to be nonlinear functions of the state variables since it's a real challenge to uncover state variables with nonlinear mapping from states to measurements [32],[33]. We generated the time series according to the following equations:where , were zero mean uncorrelated Gaussian noise with variance 0.5, 0.8 and 0.6 respectively. Hence, according to the general form of EGCM, in this toy model we have:Inspection of the above equations reveals that is a direct source to , and share a feedback loop. There is no direct connection between the remaining pairs of the state variables.

Fig. 1A is an example of the 2000 time-steps of the data and Fig. 1B shows the network structure.

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Figure 1. Results on Toy Model 1.

A. Traces of the time series. B. The causal relationships considered in Toy Model 1 between the three state variables. C. The estimated parameters , , and for the simulated data in Toy Model 1. The initial values of the three parameters are all set to 0. The covariance matrix is first set to decay slowly to achieve faster convergence and then set to decay faster after two hundred time points to ensure a better accuracy. D. Frequency decomposition of all kinds of relationships between the state variables. Significant causal influences are marked by red.

https://doi.org/10.1371/journal.pcbi.1000570.g001

The observation variables werewhere , were zero mean uncorrelated Gaussian noise with variance 0.1 and also uncorrelated with , .

Now, we can apply the method to this toy model, i.e., to estimate all the parameters and state variables from the deterministic input , and noise observations. Simulations were performed for 2 seconds (2000 equally spaced time points). Fig. 1C shows that the parameters converged to their true values with only small fluctuations after several hundred data points, even though no prior knowledge was included and the initial values of the parameters were assigned to zeros. It has already been pointed out that the covariance matrix (see Methods section) of the noise in the parameter equation will affect the convergence rate and tracking performance [34]. In the situation here, a steep decay of the covariance matrix will lead to a better accuracy but the convergence is then slow. On the other hand, a slow decay will lead to a faster convergence but a larger fluctuation is observed. Hence, was carefully controlled to reduce to zero as the increased (see Fig. 1C).

After the state variables being recovered, we computed the partial Granger causality in both time and frequency domains (see Fig. 1D) and used the bootstrap approach to construct confidence intervals. Specifically, we simulated the fitted model to generate a data set of 1000 realizations of 2000 time points and use as the confidence interval. In this result, a causal connection was illustrated as part of the network if, and only if, the lower bound of the confidence interval of the causality was greater than zero. The results show that our extended model can detect the causal relationship correctly in both time and frequency domains.

Toy Model 2.

When dealing with real data it is quite common that we need to detect the causal influence between time series from several variables affected by some stimulus. The stimulus may be very complicated, or hard to measure, and it may be impossible to formulate its form explicitly. However, if we ignore the influence of these inputs and use a traditional VAR model to detect the causality it is quite probable that we will get a misleading structure even if we use a high-order VAR model.

We used the following toy model which has exactly the same connection coefficients between the three state variables considered in Toy model 1 with an additional simple constant input function :(15)where , were zero mean uncorrelated Gaussian noise with variance 0.5, 0.8 and 0.6 respectively.

Here, we assumed that and the observation variables , were identical to the state variables with observation noise. The variance of the noise was 0.1. It is obvious that the network structure is the same one as shown in Fig. 1B. However, if we ignore the constant input and just use a VAR model to detect this structure, we obtain the structure shown in Fig. 2A and Fig. 2B with confidence intervals where two additional causal relationships (i.e. and ) are presented showing that the real causal influence can no longer be correctly detected. Furthermore, when the input is not taken into consideration, the coefficients of the connection matrix will be meaningless and no longer provide us with the correct estimation of the strength of connection strengths between the state variables. This illustrates why we consider it necessary to incorporate the stimulus into our model although sometimes we don't know its form or intensity.

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Figure 2. Results on Toy Model 2.

Network structures with and without stimulus. A. Confidence intervals of all links between units. The data is generated with Eq. (15), but we use (without input) in our algorithms and a traditional VAR(10) model to detect the causal influence. B. The network structure of the state variables corresponding to A. Two additional causal relationships are marked by the dashed line. C. Confidence intervals of all links between units. The data is generated with Eq. (16) where and , are generated with normal distribution (with input). D. The network structure of the state variables corresponding to C.

https://doi.org/10.1371/journal.pcbi.1000570.g002

With our extended model we can, to some extent, solve the above issue and detect the causal influence correctly amongst those state variables affected by some unknown stimulus intermittently, although our model is originally set up for deterministic inputs.

We next generated a time series of 10000 time points which was composed of 10 segments with equal length, a.e., . Each segment took the form:(16)where , and , were zero mean uncorrelated Gaussian noise with variance 0.5, 0.8 and 0.6 respectively.

The five segments were generated according to the above toy model without input, i.e., , while the remaining five segments were assumed to include input of random intensity which would also affect the state variables randomly. Specifically, within each segment, was assigned a random value which was generated with the normal distribution , and the same was the case with : , . Observation variables were still assumed to be identical to the state variables with the variation of observation as 0.1.

Hence, the network structure of the three state variables is still the same as shown in Fig. 1B while each state variable is affected by some input that we don't know the intensity of. Fig. 2C and Fig. 2D show the predicted network structure with confidence intervals using our extended model. The results show that we can still detect the causal influence correctly in this situation.

Comparing EGCM, GCM and DCM.

EGCM has a strong connection with DCM as well as GCM. We consider the Dynamical Causal model:(17)where are state variables and is the transpose of a vector, is a known deterministic input corresponding to designed experimental effects, is the set of parameters to estimate, is the diffusion matrix (could depend on time) and is the Brownian motion (or in general, it could be a martingale). The state variables enter a specific model to produce the outputs with the observation noise .

Here we focus on the bilinear approximation of the Dynamical Causal model which is the most parsimonious but useful form [10]:(18)whereare parameters that mediate the intrinsic coupling among states, allow the inputs to modulate the coupling, and elicit the influence of extrinsic inputs on the states respectively. Here, for simplicity, we have expanded the state equation around and assumed that .

The bilinear approximation of DCM is represented in terms of nonlinear differential equations while the GCM (see Eq. (1)) is formulated in discrete time and the dependencies among state variables are approximated by a linear mapping over time-lags which seems to be quite different. However, we can find the difference is that the bilinear form includes deterministic inputs and observation variables and equations which are not considered in GCM. The formulation (18) comes from the Volterra series and is certainly a more accurate and biophysical constraint representation of a biological system. On the other hand, the GCM with autoregressive representation always takes the past information into consideration while the bilinear approximation of DCM has no time-lags included in the differential equations although the general form of DCM may have [36]. So, if we alter the DCM to the form:(19)where is a kernel function, then the DCM shares the feature of the GCM. On the other hand, our EGCM includes both deterministic inputs and observation variables thus takes the advantages of both DCM and GCM. In the general form of EGCM (Eq. (2)), we can find that when , this is the discrete form of the bilinear form of DCM, and when , it is the GCM with additional inputs.

Advantage of extended approach.

In contrast to all previous methods in estimating Granger causality in the literature where essentially a regression method is employed, in EGCM we incorporate noise observation variables and apply the extended Kalman filter to recover the state variables. Additional inputs are also included in EGCM on the basis of an autoregressive model. The advantage of such an approach over the previous methods is obvious. The EGCM is more reasonable when we are faced with experimental data affected by a particular stimulus and applicable to cases where we cannot track the state variables respectively but just a function of them, or where the observation noise is considerable. Comparing to the traditional VAR models, all the coefficients in EGCM correspond to intrinsic or latent dynamic coupling and changes induced by each input which endow the model with interpretability power. Furthermore, all the previous methods in estimating Granger causality are batch learning: they require collection of all data before an estimation can be made. The extended Kalman filter, on the other hand, is an online learning: we can now update Granger causality instantaneously. One may argue that this is a common feature of online learning vs. batch learning. However, it is novel in the context of Granger causality. When we are faced with biological data, this feature becomes particularly significant. As we know, adaptation, or learning in animals, is very important but this makes it difficult to analyze since adaptation introduces dynamic change into the system. The classical way of estimating Granger causality can cope with this difficulty by introducing sliding windows in analyzing data. Of course, to select an optimal window size is always an issue in such an approach. However, in Kalman filtering, we can have the advantage of the connection of the state variables from the connection matrix and such an issue is automatically resolved.

In comparison with the bilinear approximation of DCM, the advantages of EGCM are the following: First, it allows time delay in the model more naturally and easier to deal with. Time delay is ubiquitous in a biological system, no matter whether we are considering gene, protein, metabolic and neuronal networks. Secondly, using Granger causality we are able to summarize the causal effect into a single number which is more transparent and easy to understand, particularly in a system with a time delay. Thirdly, it allows a frequency domain decomposition. We know that when we are dealing with a dynamic system it is sometimes much informative to view it in the frequency domain rather than in the time domain, as we have partly demonstrated here. Of course, since Eq. (19) is a continuous time version of Eq. (2), the results in the frequency domain obtained for Eq. (2) is essentially for the DCM model as well. We summarize our comparisons in Table 1 (see [10]).

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Table 1. Comparing DCM, GCM and EGCM.

https://doi.org/10.1371/journal.pcbi.1000570.t001

Other types of data.

In the current paper, we have only applied EGCM to LFP data although it is clearly applicable to many other types of biological data. For example, in gene microarray data, we can have a readout of transcriptional changes in several thousand genes at different times over a period of many hours [37]. The same is the case with multiple protein measurements over time in biological systems or in metabolic changes. In all these situations estimation of altered causal connections in both time and frequency domains will provide invaluable information about changes occurring in the relationships between different components in the systems being studied.

IT hemispheric differences and learning.

The results of the EGCM analysis of our IT LFP data provide the first evidence for connectivity changes between and within left and right ITs as a result of face recognition learning. It is clear that learning is a dynamic and complex process [38],[39]. In both sheep during learning there were more causal connections from the left to the right IT than vice-versa during learning trials. However, immediately after learning had occurred the number of left to right connections diminished to the same low level as seen from right to left. Within the hemispheres connectivity increased progressively over time in the right IT after learning but remained the same or decreased in the left IT. There was a strong negative correlation between the number of connections from left to right and the number within the right IT. This suggests that the left to right IT connections may exert some form of inhibitory control over the number within the right IT and that this therefore needs to be weakened for new face discriminations to be learned. The EGCM frequency analysis using theta and gamma oscillation data in the two hemispheres showed that during learning of new face pairs there was significantly more information being processed in the low frequency (theta) in the right IT than in the left. After learning however this declined and it appeared that the higher frequency information (gamma) became more dominant in both hemispheres. Lower frequency oscillations are more associated with global encoding over widespread areas of brain whereas higher frequencies are more associated with more localized encoding. This may suggest that during the course of learning new faces the right IT uses a more global mode of encoding to promote more rapid learning and that once learning has successfully occurred the right IT shifts to a more localized encoding strategy for maintaining learning. In the left IT on the other hand this more local encoding strategy predominates both during and after learning. This is in broad agreement with recent proposals that the left hemisphere is more involved in local encoding and the right in global encoding [18],[19] although in the case of face recognition it would appear that the right hemisphere shifts from a global to local encoding strategy once faces have been learned. Clearly more analyses of this kind are required before these differences in left and right brain hemispheres processing and interactions can be fully understood but combining multiple LFP recordings and EGCM will be a powerful future approach.

Field-type model.

In the current paper, we have not explicitly introduced the spatio-correlation between each variables (electrodes). In other words, we have ignored the geometric relationship of electrodes in the array. This is certainly an over-simplification of the real situation due to the following reasons. First of all, despite the long history of multi-electrode array recordings, in vivo recording in, for example, IT is still very rare and difficult. Even we have a reliable recording session, the obtained data set is hard to fully analyze: for example, to reliably sort the spikes [40]. Secondly, assuming we could work out the spatio-temporal model for one animal, it is almost no sense to map the results about the detailed geometrical relationship (electrodes) to another animal. Also, in our experiments, we often face the situation that we have to discard the data from quite a few electrodes. A spatio-temporal (random field) approach as developed in fMRI [41] is not considered here. However, with the improvement of experimental techniques and the introduction of new techniques (see for example, functional multineuron calcium imaging [42]), a spatio-temporal model (a random field or a random point field) is cried for and is one of our future research topics. The well developed approach: dynamic expectation maximization [43], could play a vital role here.