Log-linear model of multiple neural spike sequences.

We consider an ensemble spike pattern of neurons. The state of each neuron is represented by a binary random variable, ( where ‘’ denotes a spike occurrence and ‘’ denotes silence. An ensemble spike pattern of neurons is represented by a vector, , with the total number of possible spike patterns equal to . Let , where and or (), represent the joint probability mass function of the -tuple binary random variables, . Because of the constraint , the probabilities of all the spike patterns are specified using parameters. In information geometry [44],[50], these parameters are viewed as ‘coordinates’ that span a manifold composed of the set of all probability distributions, . In the following, we consider two coordinate systems.

The logarithm of the probability mass function can be expanded as(1)where . In this study, a prime indicates the transposition operation to a vector or matrix. is a log normalization parameter to satisfy . The log normalization parameter, , is a function of . Eq. 1 is referred to as the log-linear model. The parameters , ,…, are the natural parameters of an exponential family distribution and form the ‘-coordinates’. The -coordinates play a central role in this paper.

The other coordinates, called -coordinates, can be constructed using the following expectation parameters:(2)These parameters represent the expected rates of joint spike occurrences among the neurons indexed by the subscripts. We denote the set of expectation parameters as a vector .

Eqs. 1 and 2 can be compactly written using the following ‘multi-indices’ notation. Let be collections of all the -element subsets of a set with elements (i.e., a k-subset of elements): , , etc. We use the ‘multi-indices’ to represent the natural parameters and expectation parameters as and , respectively. Similarly, let us denote the interaction terms in Eq. 1 as(3)Using (), Eqs. 1 and 2 can now be compactly written as and , respectively.

The higher-order natural parameters in the log-linear model represent the strength of the higher-order spike interactions. Amari et al. [44], [45], [50], [51] proved that the - and -coordinates are dually ‘orthogonal’ coordinates and demonstrated that the natural parameters that are greater than or equal to the th-order, (, ), represent an excess or paucity of higher-order synchronous spikes in the () coordinates. To understand the potential influence of the higher-order natural parameters on the higher-order joint spike event rates, let us consider a log-linear model in which the parameters higher than or equal to the th-order vanish: (). In the dual representation, the higher-order joint spike event rates, (), are chance coincidences expected from the lower-order joint spike event rates, (). From this, it follows that the non-zero higher-order natural parameters that are greater than or equal to the th-order of a full log-linear model represent the excess or paucity of the higher-order joint spike event rates, (), in comparison to their chance rates expected from the lower-order joint spike event rates, ().

However, in this framework, the excess or scarce synchronous spike events of subset neurons reflected in () are caused not only by the non-zero th-order interactions (), but also by all non-zero higher-order interactions (). Therefore, one cannot extract the influence of pure th-order spike interactions () on the higher-order synchrony rates unless the parameters higher than the th-orders vanish, (). To formally extract the th-order spike interactions, let denote a log-linear model whose parameters higher than the th-order are fixed at zero. By successively adding higher-order terms into the model, we can consider a set of log-linear models forming a hierarchical structure as . Here, denotes that is a sub-manifold of in the -coordinates. In , the set of non-zero th-order natural parameters explains the excess or paucity of th-order synchronous spike events, (), in comparison to their chance occurrence rates expected from the lower-order marginal rates, (). In a full model , the last parameter in the log-linear model, , represents the pure th-order spike correlation among the observed neurons. The information geometry theory developed by Amari and others [44], [45], [50], [51] provides a framework for illustrating the duality and orthogonal relations between the - and -coordinates and singling out the higher-order correlations from these coordinates using hierarchical models. In the Methods subsection ‘Mathematical properties of log-linear model’ we describe the known properties of the log-linear model utilized in this study, and give references [44], [45], [50], [51] for further details on the information geometry approach to spike correlations.

State-space log-linear model of multiple neural spike sequences.

To model the dynamics of the spike interactions, we extend the log-linear model to a time-dependent formulation. To do so, we parameterize the natural parameters as in discrete time steps . We denote the dimension of for an th-order model as . The time-dependent log-linear model up to the -th order interactions () at time is then defined as(4)The corresponding time-varying expectation parameters at time are written as , where each element () is given as(5)

In neurophysiological experiments, neuronal responses are repeatedly obtained under identical experimental conditions (‘trials’). Thus, we here consider the parallel spike sequences repeatedly recorded simultaneously from neurons over trials and align them to a trigger event. We model these simultaneous spike sequences as time-varying, multivariate binary processes by assuming that these parallel spike sequences are discretized in time into bins of bin-width . Let be the observed -tuple binary variables in the -th bin of the -th trial, whereby a bin containing ‘’ values indicates that one or more spikes occurred in the bin whereas ‘’ values indicate that no spike occurred in the bin. We regard the observed spike pattern at time as a sample (from a total of trials) from a joint probability mass function. An efficient estimator of is the observed spike synchrony rate defined for the -th bin as(6)for . The synchrony rates up to the -th order, , constitute a sufficient statistic for the log-linear model up to the -th order interaction.

Using Eqs. 4 and 6, the likelihood of the observed parallel spike sequences is given as(7)with and . Here, we assumed that the observed spike patterns are conditionally independent across bins given the time-dependent natural parameters, and that the samples across trials are independent.

Our prior assumption about the time-dependent natural parameters is expressed by the following state equation(8)for . Matrix ( matrix) contains the first order autoregressive parameters. ( matrix) is a random vector drawn from a zero-mean multivariate normal distribution with covariance matrix ( matrix). The initial values obey , with ( matrix) being the mean and ( matrix) the covariance matrix. The parameters , , , and are called hyper-parameters. In the following, we denote the set of hyper-parameters by : .

Given the likelihood (Eq. 7) and prior distribution (Eq. 8), we aim at obtaining the posterior density(9)using Bayes' theorem. The posterior density provides us with the most likely paths of the log-liner parameters given the spike data (maximum a posteriori, MAP, estimates) as well as the uncertainty in its estimation. The posterior density depends on the choice of the hyper-parameters, . Here, the hyper-parameters, , except for the covariance matrix for the initial parameters, are optimized using the principle of maximizing the logarithm of the marginal likelihood (the denominator of Eq. 9, also referred to as the evidence),(10)

For non-Gaussian observation models such as Eq. 7, the exact calculation of Eq. 10 is difficult (but see the approximate formula in the Methods section). Instead, we use the EM algorithm [68], [78] to efficiently combine the construction of the posterior density and the optimization of the hyper-parameters under the maximum likelihood principle. Using this algorithm, we iteratively construct a posterior density with the given hyper-parameters (E-step), and then use it to optimize the hyper-parameters (M-step). To obtain the posterior density (Eq. 9) in the E-step, we develop a nonlinear recursive Bayesian filter/smoother. The filter distribution is sequentially constructed by combining the prediction distribution for time based on the state equation (Eq. 8) and the likelihood function for the spike data at time , Eq. 4. Figure 1 illustrates the recursive filtering process in model subspace . In combination with a fixed-interval smoothing algorithm, we derive the smooth posterior density (Eq. 9). The time-dependent log-linear parameters (natural parameters) are estimated as MAP estimates of the optimized smooth posterior density. In the Methods section, please refer to the subsection on ‘Bayesian estimation of dynamic spike interactions’ for the derivation of the optimization method for hyper-parameters, along with the filtering/smoothing methods. We summarize a method for estimating the dynamic spike interactions in Table 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 1. Geometric view of recursive filtering in subspace .

Each point in this figure represents a probability distribution, , of an -tuple binary variable, . The underlying time-dependent model is represented by white circles in the space of . The dashed lines indicate projections of the underlying models to the model subspace, . The maximum a posteriori (MAP) estimates of the underlying models projected on subspace were obtained recursively: Starting from the MAP estimate at time (filter estimate, red circle), the model at time is predicted based on the prior knowledge of the state transition, Eq. 8 (blue arrow, prediction; black cross, a predicted distribution). The maximum likelihood estimate (MLE, black circle) for the spike data at time derived by Eq. 4 is expected to appear near the projection point of the underlying model at time in . The filter distribution at time is obtained by correcting the prediction by the observation of data at time (black arrow). The filter estimation at time is used for predicting the model at time and so on. This recursive procedure allows us to retain past information while tracking the underlying time-dependent model based on the current observation.

https://doi.org/10.1371/journal.pcbi.1002385.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Method for estimating dynamic spike interactions.

https://doi.org/10.1371/journal.pcbi.1002385.t001

Estimation of time-varying pairwise spike interaction.

To demonstrate the utility of the developed methods for the analysis of dynamic spike correlations, we first consider a nonstationary pairwise spike correlation analysis. For this goal, we apply the state-space method to two examples of simulated spike data, with neurons. The dynamic spike correlation between two neurons can be analyzed by conceptually simpler histogram-based methods, e.g., a joint peri-stimulus time histogram (JPSTH). However, even for the pair-analysis, the proposed method can be advantageous in the following two aspects. First, the proposed method provides a credible interval (a Bayesian analogue of a confidence interval). Using the recursive Bayesian filtering/smoothing algorithm developed in the Methods section, we obtain the joint posterior density of the log-linear parameters (Eq. 9). The posterior density provides, not only the most likely path of the log-liner parameters (MAP estimates), but also the uncertainty in its estimation. The credible interval allows us to examine whether the pairwise spike correlation is statistically significant (but see the later section on “Testing spike correlation in nonstationary spike data” for the formal use of the joint posterior density for testing the existence of the spike correlation in behaviorally relevant time periods). Second, an EM algorithm developed in the proposed method optimizes the smoothness of the estimated dynamics of the pairwise correlation (i.e., optimization of the hyper-parameters, in the state equation, Eq. 8). By the automatic selection of the smoothness parameter, we can avoid the problem of spurious modulation in the estimated dynamic spike correlation caused by local noise, or excessive smoothing of the underlying modulation.

Figure 2A displays an application of our state-space method to 2 parallel spike sequences, (, , and ), which are correlated in a time-varying fashion. The data are generated as realizations from a time-dependent formulation of a full log-linear model of 2 neurons (Figure 2A left, repeated trials: ; duration: bins of width ). Here, the underlying model parameters, , , and (Figure 2A right, dashed lines), are designed so that the individual spike rates are constant ( and ), while the spike correlation between the two neurons, , varies in time, i.e., across bins (synchronous spike events caused by the time-dependent correlation are marked as blue circles in Figure 2A left). While the bin-width, , is an arbitrary value in this simulation analysis, the bin-width typically selected in spike correlation analyses is on the milli-second order. If the bin-width is , the individual spike rates of simulated neurons 1 and 2 are 38.4 Hz and 19.4 Hz, respectively. The correlation coefficient calculated from these parallel spike sequences is 0.0763. These values are within the realistic range of values obtained from experimentally recorded neuronal spike sequences. By applying the state-space method to the parallel spike sequences, we obtain the smooth posterior density of the log-linear parameters. The right panels in Figure 2A display the MAP estimates of the log-linear parameters (solid lines). The analysis reveals the time-varying pairwise interaction between the two neurons (Figure 2A right, bottom). The gray bands indicate 99% credible intervals from the posterior density, Eq. 9. We used marginal posterior densities, (), to display the credible intervals for the individual log-linear parameters. The variances of the individual marginal densities were obtained from the diagonal of a covariance matrix of the smooth joint posterior density (Eq. 35).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. Estimation of pairwise interactions in two simulated parallel spike sequences.

(A) Application of the state-space log-linear model to parallel spike sequences with time-varying spike interaction. (Left) Based on a time-dependent formulation of the log-linear model (dashed lines in the right panels represent the model parameters), parallel spike sequences, , are simulated repeatedly for trials (duration: bins). The two panels show dot displays of the spike events of the variables, or ( and ). The observed synchronous spike events across the two spike sequences within the same trials are marked by blue circles. (Right) Smoothed estimates of the log-linear parameters, (solid lines, red: pairwise interaction; blue and green: the first order), estimated from the data shown in the left panels. The gray bands indicate the 99% credible interval from the posterior density of the log-linear parameters. The dashed lines are the underlying time-dependent model parameters used for the generation of the spike sequences in the left panels. (B) Application of the state-space log-linear model to independent parallel spike sequences with time-varying spike rates. Each panel retains the same presentation format as in A.

https://doi.org/10.1371/journal.pcbi.1002385.g002

Figure 2B shows an application of the method to parallel spike sequences of time-varying spike rates. Here, the underlying model parameters are constructed so that the two parallel spike sequences are independent ( for ), while the individual spike rates vary in time (i.e., across the bins). The observation of synchronous spike events (Figure 2B left, blue circles) confirms that chance spike coincidences frequently and trivially occur at higher spike rates. The analysis based on our state-space method reveals that virtually no spike correlation exists between the two neurons, despite the presence of time-varying rates of synchronous spike events (Figure 2B right, bottom).

Simultaneous estimation of time-varying pairwise spike interactions.

In this subsection, we extend the pairwise correlation analysis of 2 neurons to the simultaneous analysis of multiple pairwise interactions in the parallel spike sequences obtained from more than 2 neurons.

Figure 3 demonstrates an application of our method to simulated spike sequences of neurons. As an underlying model, we construct a time-dependent log-linear model of 8 neurons with time-varying rates and pairwise interactions (, , duration: bins). The higher-order log-linear parameters are set to zero, i.e., no higher-order interactions are included in the model. Figure 3A displays snapshots of the dynamics of the parameters of the individual spike rates, (), and pairwise interactions, (), at . Figure 3B shows the parallel spike sequences (50 out of 200 trials are displayed) simulated on the basis of this model. The spikes involved in the pairwise, synchronous spike events between any two of the neurons (in total: 28 pairs) are superimposed and marked with blue circles. Figure 3C displays snapshots of the simultaneous MAP estimates of the pairwise interactions, (), of a log-linear model of 8 neurons applied to the parallel spike train data. In addition, the spike rates were estimated from the dual coordinates, i.e., (). The results demonstrate that the simultaneous estimation of time-varying, multiple pairwise interactions can be carried out by using a state-space log-linear model with up to pairwise interaction terms.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Simultaneous estimation of pairwise interactions of 8 simulated neurons.

(A) Snapshots of the underlying model parameters of a time-dependent log-linear model of neurons containing up to pairwise interactions (duration: bins) at bins. No higher-order interactions are included in the model. Each node represents a single neuron. The strength of a pairwise interaction between the th and th neurons, (), is expressed by the color as well as the thickness of the link between the neurons (see legend at the right of panel B). A red solid line indicates a positive pairwise interaction, whereas a blue dashed line represents a negative pairwise interaction. The underlying spike rates of the individual neurons, (), are coded by the color of the nodes (see color bar to the right of panel A). (B) Dot displays of the simulated parallel spike sequences of 8 neurons, , sampled repeatedly for trials from the time-dependent log-linear model shown in A. For better visibility, only the first trials are displayed (). Synchronous spike events between any two neurons (28 pairs in total) are marked by blue circles. (C) Pairwise analysis of the data illustrated in B (using all trials) assuming a pairwise model () of 8 neurons. The snapshots at the bins show smoothed estimates of the time-varying pairwise interactions, (), and the spike rates, (). For this estimation, we use for the prior density of initial parameters. The scales are identical to the one in panel A.

https://doi.org/10.1371/journal.pcbi.1002385.g003

Estimation of time-varying triple-wise spike interaction.

Another important aspect of the proposed method is its ability to estimate time-varying higher-order spike interactions that cannot be revealed by a pairwise analysis. To demonstrate this, we apply the state-space log-linear model to parallel spike sequences by considering up to a triple-wise interaction (i.e., the full log-linear model). Spike data (Figure 4A) are generated by a time-dependent log-linear model (Figure 4C, dashed lines) repeatedly in trials. Figure 4C displays the MAP estimates (solid lines) of the log-linear parameters from the data shown in Figure 4A. Here, non-zero parameter represents a triple-wise spike correlation, i.e., excess synchronous spikes across the three neurons or absence of such synchrony compared to the expectation if assuming pairwise correlations. The gray band is the 99% credible interval from the marginal posterior density, , for .

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. Estimation of triple-wise interaction from simulated parallel spike sequences of 3 neurons.

(A) Dot displays of the simulated spike sequences, , which are sampled repeatedly for trials from a time-dependent log-linear model containing time varying pairwise and triple-wise interactions (duration: bins; see the dashed lines in C for the model parameters). Each of the 3 panels shows the spike events for each of the 3 variables, ( and ), as black dots. Synchronous spike events across the 3 neurons as detected in individual trials are marked by blue circles. (B) Observed rates of joint spike events, (). (Top) Observed rates of the synchronous spike events between all possible pair constellations as specified by index (). (Bottom) Observed rate of the synchronous spikes across all 3 neurons, . (C) Smoothed estimates of the time-varying log-linear parameters, . The three panels depict the smoothed estimates (solid lines) of the log-linear parameters, , of the different orders (), as obtained from the data shown in A and B (top and middle: the first and second order log-linear parameters; bottom: triple-wise spike interaction, ). The gray bands indicate the 99% credible interval of the marginal posterior densities of the log-linear parameters. The dashed lines indicate the underlying time-dependent parameters used for the generation of the spike sequences.

https://doi.org/10.1371/journal.pcbi.1002385.g004

The credible interval of the higher-order (triple-wise) log-linear parameter in the bottom panel of Figure 4C appears to be larger than those of the lower-order parameters. In general, the observed frequency of simultaneous spike occurrences decreases as the number of neurons that join the synchronous spiking activities increases (note that the marginal joint spike occurrence rate, Eq. 2, is a non-increasing function with respect to the order of interaction, i.e., if the elements of are included in ). Thus, the estimation variance typically increases for the higher-order parameters, as seen in the bottom panel of Figure 4C. Related to the above, because of the paucity of samples for higher-order joint spike events, the automatic smoothness optimization method selects hyper-parameters that make the trajectories of the higher-order log-linear parameters stiff in order to avoid statistical fluctuation caused by a local noise structure. Given the limited number of trials available for data analyses, these observations show the necessity of a method to validate inclusion of the higher-order interaction terms in the model.

Additionally, we observe that in the later period of spike data (300–500 bins), the dynamics of the estimated triple-wise spike interaction do not follow the underlying trajectory faithfully: The underlying trajectory falls on outside the 99% credible interval. Similar results are sometimes observed when an autoregressive parameter, , in a state model is optimized (Eq. 8). In contrast, when we replace the autoregressive parameter with an identity matrix (i.e., , where is the identity matrix), the credible intervals become larger. Therefore, such observations do not typically occur. Thus, we also need a method for validating the inclusion of the autoregressive parameter in the state model using an objective criterion. Detailed analyses of these topics will be given in the next section using the example of 3 simulated neurons displayed in Figure 4. In the above example, the state-space log-linear model with an optimized provides a better overall fits to the spike data than the model using , despite an inaccurate representation in part of its estimation. However, for the purpose of testing the spike correlation in a particular period of spike data, we recommend using an identity matrix as an autoregressive parameter, i.e. , in the state model.

Selection from hierarchical models.

Here, we examine the validity of including higher-order interaction terms in the model by using the AIC. We apply the model selection method to the spike train data of simulated neurons. The data are generated by the time-varying, full log-linear model that contains a non-zero triple-wise interaction terms shown in Figure 4C (dashed lines). The AICs are computed for hierarchical state-space log-linear models, i.e., for models of interaction orders up to . To test the influence of the data sample size on the model selection, we vary the number of trials, , used to fit the hierarchical log-linear models. The results are shown in Table 2 for . For a small number of trials (), a model without any interaction structure () is selected. For larger numbers of trials, models with larger interaction orders are selected. For , the full log-linear model () is selected.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. AICs for different numbers of trials.

https://doi.org/10.1371/journal.pcbi.1002385.t002

Below, we examine whether the AIC selected a model that minimizes the KL risk function by directly computing its approximation using the known underlying model parameters. First, Table 3 shows how often a specific order is selected by the AIC by repeatedly applying the method to different samples generated from the same underlying log-linear parameters (Figure 4C, dashed lines). We examine two examples: One in which a sample is composed of trials (left) and the other of trials (right). We repeatedly compute the AICs of state-space models of different orders () applied to 100 data realizations (of the respective number of trials). We then count how often a model of order is selected by minimizing the AIC. For comparison, the table includes the outcomes from other criteria such as the Bayesian information criterion (BIC) [79], [80] and the predictive divergence for indirect observation models (PDIO) [81], which are suggested for models containing latent variables. Please see the Methods section for the details of these criteria. Next, Table 4 displays the most frequently selected model (from  = 1,2,3) by the various information criteria when they are applied to 100 data realizations as a function of the number of trials in each data set, (see Table 3 for the outcomes of and ).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Models selected using different information criteria.

https://doi.org/10.1371/journal.pcbi.1002385.t003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Model orders selected by different information criteria for different numbers of trials.

https://doi.org/10.1371/journal.pcbi.1002385.t004

We compare the outcomes of the AIC and two other information criteria with the model order that minimizes the KL risk function (KL-risk). For this goal, we include in Table 4 the model order that minimizes the KL-risk between the underlying log-linear model (Figure 4C, dashed lines) and estimates of the th-order model. In addition to the KL-risk, we calculate the mean squared error (MSE) between the underlying model parameters and corresponding estimates. Please see in the caption of Table 4 how we compute the KL-risk and MSE for this analysis. We find that the KL-risk and MSE select the same model, except for the case of trials. In comparison to the KL-risk, the BIC tends to select models with an excessively higher-order of interaction (over-fitting) for a small number of trials () and tends to choose lower-order models for . The PDIO mostly selects lower-order models. In contrast, the AIC follows the selection of the KL-risk minimization principle, except for , where it shows a conservative choice compared to the KL-risk.

We repeat the same analysis for spike data generated from an underlying model that contains up to pairwise interactions, but does not contain the triple-wise interaction term. The purpose of this analysis is to show that the methods do not select models with excessively higher orders of interaction than those actually contained in the data. To construct such an underlying model, we project the full model (shown in Figure 4C) onto the subspace of a pairwise log-linear model, . The projection model does not contain any triple-wise correlation, while the 1st and 2nd order expectation parameters ( for ) are the same as those of the full model that was used to generate the data in the analysis of Tables 2–4. Table 5 displays the most frequently selected model orders by the AIC, BIC, PDIO, along with the selections by the KL-risk and MSE. We find that the pairwise model is the most frequently selected model under all of the criteria for the samples with a large number of trials (). Under this condition, only the AIC among the other data-driven methods follows the KL-risk selection. These results lead us to the conclusion that the AIC is a reliable measure to assess the goodness of fit of the state-space log-linear model.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Model orders selected using different information criteria for model without triple-wise spike interaction.

https://doi.org/10.1371/journal.pcbi.1002385.t005

Selection of state transition model.

In addition to validating the inclusion of the order of spike interactions, we examine state models (Eq. 8) with different conditions for the hyper-parameters by the AIC, using as an example the same spike train data of 3 neurons with trials (displayed in Figure 4A). The tested state models are (i) a time-independent model in which the hyper-parameters are fixed as and , where is an identity matrix; (ii) a random walk model (), in which only the covariance matrix, , is optimized via the EM algorithm; and (iii) a 1st-order autoregressive model, in which and are both optimized. With these settings, the state-space log-linear model of case (i) becomes a stationary log-linear model, which has been frequently employed in spike data analyses, often in the form of a maximum entropy model [55]–[57], [82]. In contrast, the state-space models of cases (ii) and (iii) are nonstationary because the joint distribution of the spike pattern changes in time as a result of the time-varying log-linear parameters. The interaction parameters themselves are assumed to be nonstationary in the state model in case (ii), and can be either stationary or nonstationary in the state model in case (iii). To see this, let be the eigenvalues of , i.e., the solutions for . For a stationary process, the eigenvalues have to satisfy , otherwise the process is nonstationary. This excludes the case .

The AICs of the full model with the various state-equations become progressively smaller for increasingly complex state-space models (38525 for case (i), 36263 for case (ii), and 36231 for case (iii)). The fact that conditions (ii) and (iii) are better fitted to the data than condition (i) confirms that the proposed method of time-varying spike correlation analysis performs better than an analysis based on a stationary log-linear model for this data. In addition, the fact that condition (iii) better fits the data than condition (ii) supports the use of autoregressive models (note: The estimation in Figure 4B is done with state model (iii)). We find that the fitted (to the data in Figure 4A) yields an eigenvalue larger than 1, indicating that the underlying state process ( bins) is modeled as a nonstationary process. We consider that the nonstationary state model is selected because of the relatively short observation period, during which a nonstationary trend for the parameters can appear even if the state process is stationary in the long run.

Detecting triple-wise spike correlation in simulated nonstationary spike sequences.

In this subsection, we examine a method to test for the presence of a higher-order (triple-wise) spike correlation using simulated spike train data with known spike interaction dynamics. In this context, we again use the simulated spike data of neurons (of length 500 bins with trials) shown in Figure 4A, which was generated by the time-varying model, as shown in Figure 4C (dashed lines). In the following, we assume for this data an underlying experimental protocol that can be segmented into behaviorally-relevant time periods, I–IV, as indicated in Figure 5A, as for example a period before the trial starts, a preparatory period, a movement period, etc. (e.g., see [8], [19] for a typical experimental protocol).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. Detection of spike correlations and their relation to pseudo experimental protocol (simulation study).

(A) Sketch of an assumed experimental time course composed of four epochs (I–IV), e.g., of different behavioral task conditions. Epochs I–III have a duration of 100 bins, period IV has a duration of 200 bins. (B) Time-varying triple-wise spike interaction parameter of the underlying model (cf. Figure 4C, ) used for the simulation of spike data ( neurons, trials) during the time course outlined in A. The gray areas indicate the time intervals in which the triple-wise interaction is positive: . (C) Hypothesis testing for a triple-wise spike correlation based on surrogates. In each time period (I–IV), we perform a test on the Bayes factor (BF) resulting from the original data. The observed BF (Eq. 12), marked by a red line and triangle, is computed as evidence of a positive triple-wise interaction, : , as opposed to a zero or negative triple-wise interaction, : . We then compare the ‘observed BF’ with cumulative distribution functions (CDFs, solid lines) for the BFs derived from surrogate data sets generated from a model containing only up to pairwise interactions (). In all of the CDFs, the gray area indicates the 95% confidence interval of the distribution. If the observed BF falls into the lower tail of the distribution (blue area), is supported; if it falls into the upper tail (red marked area), is supported. (D) Time courses of the underlying pair-interaction parameters of a pairwise log-linear model (), (). These underlying parameters are obtained by projecting the full underlying model () in Figure 4 to the pairwise model space, . The gray areas indicate the time periods in which all of the pairwise interactions are positive. (E) Similar tests as in C, but for the BF computed as evidence for the presence of simultaneous positive pairwise interactions, , as opposed to the absence of such an assembly, . (F) Compact visualization of the test results from C and E. The colored bars show which hypotheses are supported in the different time segments (red and blue) and where the null hypothesis cannot be rejected.

https://doi.org/10.1371/journal.pcbi.1002385.g005

In each of the time periods, we compare opposing models for the hidden log-linear parameters of neurons, using a full model (). In one hypothetical model, , we assume that the triple-wise interaction term is positive, . The time (bin) index expresses that the BF is computed under the same models for all of the time steps, , in the respective time period. In the other model , we assume that the triple-wise interaction term is smaller than or equal to zero, . Neither model makes specific assumptions about the first and second order log-linear parameters; thus, they are allowed to be real numbers. These parameters are integrated out in Eq. 13. In this simulation study, we independently applied a state-space log-linear model to each of the four periods and computed the respective BFs. Figure 5B displays the ground truth of the time-dependent triple-wise interaction parameter, , of the full log-linear model. The gray areas indicate periods where the triple-wise spike interaction, , is positive: Model is true in this period. In the remaining periods (white), the model of a negative triple-wise spike interaction, , is true.

We compare the observed BF for the tested correlation models to the BFs computed from surrogate data sets resampled under a null hypothesis of no tested order of spike interaction. Specifically, to test the existence of a triple-wise spike correlation, the surrogate BFs are computed from resampled spikes generated from a model containing no triple-wise spike interaction. For this, we first apply a pairwise state-space log-linear model () to the observed spike data to derive estimates of the time-varying spike rates and dynamic pair-interactions. Then we generate 1000 surrogate samples of parallel spike sequences, where each sample is composed of trials, using the fitted pairwise state-space log-linear model. Thus, the resampled spike sequences exhibit the same time-varying spike rates and pairwise correlations as the original data, but the triplet coincidences occur on a chance level, as expected by the individual spike rates and pairwise correlations among the neurons. The surrogate BFs of a triple-wise spike correlation (using , ) are computed for each surrogate data set by applying the full model ().

Figure 5C shows the observed BF (red vertical line and red triangle) and the cumulative distribution function (CDF) of the surrogate BFs (solid black lines) for each of the four time periods, I–IV. In each of the graphs, the gray area indicates the 95% confidence interval derived from the distribution of 1000 respective surrogate BFs. If the observed BF falls outside the confidence interval, the null hypothesis that no tested order of spike interaction exists is rejected. Further, by the two-tailed test, if the observed BF falls outside the confidence interval into the red area of the distribution on the right (i.e., significantly positive BF values), the data supports model as an underlying correlation structure, whereas if it falls into the blue area on the left, we conclude that it is a significantly negative BF, i.e., is supported. If the observed BF falls into the 95% confidence interval, the null hypothesis that no tested order of spike correlation exists is not rejected. We first look at the results for periods II and IV in Figure 5C. In these periods, the null hypothesis of no triple-wise spike correlation is rejected, and the observed BF correctly suggests a positive triple-wise correlation model (), reflecting the fact that the underlying triple-wise interaction term is positive throughout the two periods (see periods II and IV in Figure 5B). In contrast, in periods I and III, where a negative triple-wise correlation model () is true, the null hypothesis of no triple-wise correlation cannot be rejected.

It is possible that the synchronous spike events observed among multiple neurons are detected as simultaneous increases in pair interaction terms using the log-linear model of the second order (), without having to take higher-order spike interactions into consideration. Paradoxically, this issue becomes relevant when a triple-wise spike correlation is a dominant factor in the generation of synchronous spike events because the projection of a full model () that contains a positive triple-wise interaction term to a pairwise model induces simultaneous increases in the pair interaction terms in that projected model (). In such a case, an analysis of the higher-order spike correlations might just add redundant information about an animal's behavior to lower-order analyses. Thus, we repeat an analysis similar to the one above under this alternative hypothesis, using a model that contains up to pairwise interactions (). In this test, the BF (Eq. 12) is computed using the following assumed models on the hidden log-linear parameters: , all three neurons simultaneously exhibit positive pairwise interactions ( for all ); and , at least one of the pairwise interactions is not positive ( for at least one ). The first order log-linear parameters are again assumed to be real values. Surrogate data sets are obtained by resampling spike sequences from the first order state-space log-linear model () fitted to the data. The surrogate BFs for the model of simultaneously positive pairwise interactions, as opposed to its complementary model, are computed for each of the surrogate data sets by applying a pairwise model (). Figure 5D displays the ground truth of the dynamics of the pairwise interaction terms, (), of the pairwise log-linear model. These are obtained by projecting the full underlying model (Figure 4C dashed lines) to the probability space of the second-order model (). Note that in period II, the pair-interaction terms simultaneously increase because the effect of the triple-wise interaction in is projected to this sub-space model () (see Figure 4C, middle panel, for comparison, and observe that there is no such increase in the pair-interaction terms of the full log-linear model). The gray areas indicate the period where model (simultaneously positive pairwise interactions) is true. Figure 5E displays the results for testing the simultaneous increases in the pair interactions. We again look at periods II and IV. In period II, the null hypothesis of independent spike sequences is rejected, and model is supported, reflecting the fact that the projected pairwise interactions are positive for most of this time period. In contrast, the same null hypothesis cannot be rejected in period IV because neither model nor alone can support the data in this period: Models and are true in the first and second halves of period IV, respectively (see Figure 5D).

Figure 5F summarizes the results for periods I–IV using bars in corresponding colors. The results of our tests reflect the dynamics of the underlying parameters in Figures 5B and D, and we find that in period IV, only the test for a triple-wise spike correlation detected the presence of interactions among all three neurons in the data. This is because, in this time period, the dynamics of the underlying triple-wise interaction correlate well with the assumed behavioral time table, whereas the dynamics of the simultaneous pairwise interactions do not. In summary, it is possible that the higher-order analysis allows us to discover the correlated activities of multiple neurons associated with behavioral events, which may not be revealed by pairwise analyses.

Detecting triple-wise spike correlation in neural spike data from behaving monkey.

Finally, we demonstrate the application of our method on simultaneous spike recordings from the primary motor cortex (MI) of an awake, behaving monkey. The data were recorded by Alexa Riehle and her colleagues [8] to test the hypothesis that neuronal cooperativity is involved in the planning of motor actions. Therefore, Riehle et al. designed a behavioral task in which different durations of preparation intervals were provided to the monkeys before they had to perform an arm movement and touch a target on a screen. The detailed time table of the task is as follows (cf. Figure 6A). A trial started by a signal indicating to the monkey that he may initiate the task by pressing a button. After initiating the trial, the monkey had to wait for 1000 ms until a preparatory signal (PS) was delivered, indicating to the monkey that now the preparation interval started. After appearance of a response signal (RS) at 600, 900, 1200, or 1500 ms (randomly selected with equal probability), the monkey had to move his arm and touch the target position. The reaction times of the monkey showed a dependence on the duration of the preparatory period (PP): The longer the PPs the shorter the reaction times [8], [19], [84]. The conditional probability for the occurrence of the RS increases during the longer PP, in particular at the times when the RS may occur. The hypothesis for the underlying network activity realizing reduced reaction times with longer PPs was that the system is better prepared for the requested movement by increasingly enhancing the synchrony in the relevant network to facilitate the response. Indeed, Riehle et al. showed that cortical neurons modulate the degree of excess synchronous spiking activities independently from their individual firing rates in conjunction with the occurrence of the expected RSs [8], [19], [84]. An open question that could not be conclusively answered in that study was if larger groups of neurons than pairs coordinate their activity in relation to the expected events. Here, we approach this question by applying our newly developed method and will indeed demonstrate the existence of higher-order (triple-wise) spike interaction in relation to the behavioral task.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Analysis of experimental spike data using the state-space log-linear model.

(A) Experimental time table of delayed-response hand movement task. The experiments were designed and conducted by Riehle and her colleagues [8], [84]. During the preparatory period (PP, 1500 ms) that starts with the preparatory signal (PS), the presentation of the response signal (RS) was expected at three distinct moments at 600, 900, and 1200 ms (expected signals, ESs). Here the RS occurs finally after the longest possible delay of 1500 ms. After the RS, the requested movement was executed (reaction and movement time, RT-MT). See [8], [84] and the text for the detailed experimental protocol. (B) Dot displays of the spike sequences (duration: 2 s, sampling resolution 1 ms) of three neurons simultaneously recorded from the primary motor cortex (MI). The spike sequences are aligned at the onset of the PS ( trials). The synchronous spike events across the 3 neurons detected in individual trials (detection in bins of width) are marked by blue circles. (C) Observed rates of joint spike events, (). All of the events are detected in bins with a width of . (Top) Observed rates of the spike occurrence of the individual neurons (). (Middle) Observed rates of the synchronous spike events between two neurons specified by index (). (Bottom) Observed rate of the synchronous spike events of all 3 neurons, . (D) Estimation of the time-varying log-linear parameters of 3 neurons, (), from the binary data shown in C, according to the method summarized in Table 1. The panels depict the MAP estimates of the log-linear interaction parameters (solid lines; from top to bottom, the first and second order log-linear parameters and a triple-wise spike interaction, ). The gray bands indicate the 95% credible interval. In this analysis, we used an identity matrix as an autoregressive parameter, i.e. , in the state model. The covariance matrix of the initial parameter was fixed to . (E) The smoothed estimate of a triple-wise spike interaction, , is computed from binary data constructed using a bin-width of 2 ms (Top) and a bin-width of 5 ms (Bottom). The top of each panel shows the timing of the synchronous spike events of all three neurons.

https://doi.org/10.1371/journal.pcbi.1002385.g006

To that end, we analyze data which were in part analyzed by the Unitary Events analysis method (neuron id 2 and 3; shown in Figure 2 in [8]). Figure 6B shows this particular data set that consists of three simultaneously recorded neurons (neuron id 2, 3, and 5; Here we denote them as neuron 1, 2, and 3) observed during trials () of the longest preparatory period of 1500 ms. As in Riehle et al. (Figure 2 in [8]) we align the trials at the PS, and analyze the data for the interaction parameters as a function of time from 200 ms before the PS and 1800 ms after the PS. This time segment is composed of three behavioral epochs: an interval before the PS (200 ms), the PP with three expected signals (ESs, at 600, 900 and 1200 ms), and a period after the RS at 1500 ms including the reaction time and partly the movement (reaction and movement time, RT-MT) (see Figure 6A). The average firing rates of the neurons during the PP are 29.4, 12.9, and 41.9 Hz for neurons 1, 2, and 3 (neuron id 2, 3, and 5) respectively. We constructed binary sequences, , from the spike times of the neurons by binning the data using a bin-width of . Figure 6C displays the occurrence rates for the individual spiking activities (upper panel) and the pair and triple synchronous spike events (middle and bottom panels) detected in the bins with a width of .

We apply a full log-linear model of neurons with up to a triple-wise interaction term to the binary spike data shown in Figure 6C. Figure 6D displays the estimated dynamics of the interaction parameters in the log-linear model by using the method summarized in Table 1. The thick bold lines indicate the MAP estimates, i.e., the most probable paths, of the interaction parameters. The gray bands and thin solid lines mark the 95% credible interval (the Bayesian analogue of the confidence interval) computed from a marginal posterior density. The last parameter of the log-linear model, , indicates the triple-wise interaction among the three neurons in the MI. Strong positive (or negative) value of this term indicates that the three neurons are triple-wise correlated, i.e. dependent in a manner that cannot be explained by pair correlations of the neurons. Specifically, a positive triple-wise interaction value means that the occurrence of the synchronous events among the three neurons is more frequent than the chance level expected from the observed individual rates and pairwise spike correlations among them.

The analysis of the MI neurons using the state-space log-linear model reveals that the triple-wise interaction, , during the PP gradually increases, with additional local peaks at the ESs that also increase in heights towards the end of the PP (Figure 6D, bottom panel). This result is consistent with the results found by Riehle and collaborators for a subset of the neurons of the same data set (Figure 2 in [8]). In this previous study neuron 1 and 2 (neuron id 2, 3) were analyzed for excess spike synchrony using the Unitary Events analysis [32], [33]. The analysis revealed that the two neurons exhibit a modulation of significant excess spike synchrony, with peaks at the ESs (despite the first) and at the RS.[8], [32], [84]. That observation was interpreted as evidence that the neurons cooperate to prepare for motor action and facilitate the efficiency for movement execution [8], [84]. Similarly, occurrences of synchronous spike events of the three neurons roughly coincide with the expected events (ESs) (Figure 6C, bottom panel); accordingly, our result shows that the triple-wise interaction, , is also locked to the ESs, however decreases at RS. Note that the triple-wise interaction is not only determined by the frequency of synchronous spike events of all three neurons but is also determined by frequencies of other observed spike patterns across trials. However, for typical neural spike train data with low spike rates and low rates of synchronous spiking of pairs of neurons, the occurrence of synchronous spike events of three neurons can significantly increase the triple-wise interaction. In the subsection ‘Bayesian model comparison method for detecting spike correlation’ in the Methods section we explore the contribution of different spike patterns in different spiking scenarios to the evidence of a triple-wise spike correlation in simulated data. Figure 6E displays modulation of the interaction term, , using the binary data constructed with bins of smaller ( ms, upper panel) and larger (, lower panel) widths. The top of each panel shows the timing of the synchronous spike events of all three neurons. The sample sizes for synchronous spike events across all three neurons for the 2 ms bin-width are greatly reduced from those observed for a bin-width of 3 ms. Thus estimated dynamics is much less structured: The data prevents us from detecting existence of a triple-wise spike correlation using the proposed statistical test (see below for the application of the test method). With larger bins of a width of 5 ms, the precise locking of the synchronous spike events across the three neurons to ESs is no longer apparent. However, the state-space log-linear model reveals a gradual increase in the strength of the triple-wise interaction until the end of the PP.

To strengthen the findings that a triple-wise interaction term increases during the PP, we test for the presence of a triple-wise spike correlation in the PP using the Bayes factor (marginal likelihood ratio, Eq. 12). We compute the BF for the opposing models using a full model () of neurons for the hidden log-linear parameters. In one hypothetical model, , we assume that the triple-wise interaction term is positive, , in the other model, we assume that the triple-wise interaction term is smaller than or equal to zero, . A large positive value of the BF indicates that the spike data in that period supports model as opposed to model , i.e., the existence of excess synchronous spike events among the three neurons in comparison with the chance rate given by spike rates and pairwise correlations. A large negative value of the BF shows support of model as opposed to model , i.e., a paucity of synchronous spikes for the three neurons. Figure 7A shows the BF computed bin-by-bin (Eq. 13) in bit units. In the bottom panel of Figure 7A, we indicate the behavioral periods for which we test for the presence of triple-wise spike correlation (the PP and RT to MT; in later analysis we divide the PP into early and late stages). The evidence for model , as opposed to , in the behaviorally relevant time periods is obtained by summing the log of the bin-by-bin BF in that period (cf. Eqs. 12 and 13). The BF in the PP is found to be 18.08 bit, which is interpreted as ‘very strong’ evidence for presence of a positive triple-wise spike correlation by the classical guideline [76].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 7. Detection of triple-wise spike correlation of MI neurons.

(A) (Top) The bin-by-bin Bayes factor (BF) for a model of triple-wise spike interaction computed locally in time in bit units, Eq. 13. The bin-by-bin BF is computed from Eq. 13, using the state-space log-linear model fitted to the spike data (a total of 36 trials, 2 s binned using 3 ms bin-width; cf. Figure 6C). The BF computes evidence of a positive triple-wise spike interaction, , as opposed to a zero or negative triple-wise interaction, . The evidence for model as opposed to in a behaviorally relevant time period is obtained by summing the log of the bin-by-bin BF in that period (cf. Eq. 12). (Bottom) Two behavioral periods (preparatory period, PP, and reaction and movement time, RT-MT) tested for presence of a triple-wise spike correlation. In addition, to examine the evolution of the triple-wise spike correlation in the PP, the PP is divided into early and late stages at the middle of the PP. (B) (Left) The observed BF for an entire PP (marked by a red line and triangle) computed using Eq. 12 (bin-width: 3 ms). We then test the ‘observed BF’ using a distribution function (solid line) of null BFs derived from surrogate data sets generated from a fitted model containing only up to pairwise interaction terms (). The gray area indicates the 95% confidence interval of the distribution. Vertical dashed lines indicate the 90% confidence interval. If the observed BF falls into the lower tail of the distribution (blue area), is supported, if it falls into the upper tail (red marked area), is supported. (Right) The same analysis in the left panel, but with binary data constructed using larger (5 ms, upper panel) and smaller (2 ms, lower panel) bin-widths. (C) (Top) Test of observed BFs (bin-width 3 ms) computed in distinct periods: early PP (0–750 ms), late PP (750–1500 ms), and RT-MT (1500–1800 ms). (Bottom) The same test as in the top panel, except that binary data using a bin-width of 5 ms were used. (D) Test of observed BFs (bin-width: 3 ms) in each period using spike data from only the last half of the 36 trials (trials 19–36).

https://doi.org/10.1371/journal.pcbi.1002385.g007

Next, we test the observed value of the BF in the PP (18.08 bit) by comparing it with surrogate BFs. These are computed from null data that contain no triple-wise spike interactions, while keeping the time-varying structure of the pairwise correlations and the individual spike rates the same as those observed in the original spike train data. The construction of the null data follows the same procedure as in the simulation study shown before: We apply a pairwise state-space log-linear model () to the spike data and then generate 1000 surrogate samples (each with trials) of parallel spike sequences using the fitted pairwise state-space log-linear model. Figure 7B (left panel) displays the surrogate distribution along with the observed BF in the PP. The observed positive BF falls out of the 95% confidence interval, suggesting the presence of a positive triple-wise interaction as an underlying model for the spike train data in this period. In the right panels of Figure 7B, we display the results of the same analysis, but using spike train data binned by a larger (, top panel) and a smaller (, bottom panel) bin-width. We obtained almost the same results as in the analysis with a bin-width of 3 ms as for the analysis with the larger bin-width (5 ms). However, we could not reach the same results for the smaller bin width (2 ms) because the small count of synchronous spike events made it impossible to reject the null hypothesis.

If the higher-order dependency among the three neurons is related to motion preparation, the evidence for the triple-wise spike correlation should be stronger in the late stage of the PP than before. To test this idea, we divided the PP into earlier and later stages of the PP (each with a 750 ms duration, see the bottom of Figure 7A), and investigate whether the three MI neurons exhibit a triple-wise spike correlation in these periods. In addition, we select a duration of 300 ms after the onset of the RS, during which the animal starts to initiate the motor action (reaction and movement time, RT-MT). The upper panels of Figure 7C display the results obtained with bins of width ms. The weights of evidence for the triple-wise spike correlation models are 7.16, 11.2, and 0.98 bit in the early PP, late PP, and RT-MT periods respectively: i.e., ‘very strong’ evidence is found at the late stage of the PP. The null hypothesis is rejected in this late period of the PP whereas the same null hypothesis is not rejected in the earlier period of PP or the period after the RS. In the late PP, the significantly large positive BF indicates the existence of a positive triple-wise spike correlation. The bottom panels of Figure 7C display the same analyses, but with a larger () bin-width. We obtain the same results as in the analysis with a bin-width of 3 ms. When we analyze the data using a smaller bin-width (), the BFs are not significant in any of the periods (not shown here) because of the small samples of synchronous spikes. In addition, as in the simulation study in the previous subsection, we tested whether the observed synchronous spiking activities are explained merely by simultaneous increases in the pairwise interaction terms of the second order log-linear model. We did not find any evidence for such simultaneous increases in the pairwise interactions from the data for all three periods (bin-width ) (not shown here). Thus only by the application of the higher-order analysis we were able to detect the task-dependent changes in the joint interactions of all three neurons.

The critical assumptions made in this study are independence and identical sampling across the trials (across-trial stationarity). However, in the spike sequences of neuron 2 (neuron id 3) shown in Figure 6B, we notice an increase in the firing rates across trials. Higher firing rates yield a higher chance rate for synchronous spiking events. Thus, underestimation of the spike rate caused by averaging across the entire trials might induce spurious estimation of higher-order spike interaction. Indeed, synchronous spike events of the three neurons are mostly observed in later trials (e.g., trials 19–36, see blue dots in Figure 6B) when the firing rates were higher than in the earlier trials. Hence, we repeat the same analysis by using only the latter half of the trials (trials 19–36). Figure 7D displays the results. We still observe a significantly positive BF in the late PP. We note that the estimated dynamics of the log-linear parameters using only the latter half of the trials are not changed much from those using all trials, i.e., Figure 6D. In the analysis using the first half of the trials (1–19, not shown), the BF is not significant during the late PP. Kilavik et al. [19] reported that “during practice, the temporal structure of synchrony was shaped, with synchrony becoming stronger and more localized in time during late experimental sessions”. We observe similar effects in our triple-wise spike correlation analysis.

E-step: Bayesian recursive filter/smoother.

The E-step is composed of filtering and smoothing algorithms conducted by forward and backward recursions, respectively. The forward algorithm sequentially constructs a posterior density of the state at time given the spike data up to and including time , whereas the backward algorithm constructs a posterior density at time given the entire data. The posterior density allows us to compute the maximum a posteriori (MAP) estimate or Bayes estimator and provides uncertainty for the estimate. In the following, and denote the conditional mean, , and covariance, . The filter mean and covariance are denoted as and , respectively. The mean and covariance of a smooth posterior density are denoted as and , respectively.

We first compute the one-step prediction density, . This is the conditional density of the state at time given the observation of parallel spike sequences up to time . The one-step prediction density is written using the Chapman-Kolmogorov equation as [37], [69], [105](24)Here the transition probability, , is a multivariate normal distribution with mean and covariance , as defined in the state equation, Eq. 8. For an initial prior, , we use a normal distribution with mean and covariance . The other distribution, , in Eq. 24 is the filter density at time . In the next paragraph, the filter density will be obtained by approximating it with a normal distribution whose mean and covariance are denoted as and . Under this condition, the one-step prediction density (Eq. 24) again becomes a normal distribution whose mean, , and covariance, , are given as [37], [68](25)(26)

The filter density, , is the conditional distribution of the state given the observation of parallel spike sequences up to time . Using the likelihood function and one-step prediction density, the filter density is given by Bayes' theorem as(27)This posterior density is a complicated function with respect to the natural parameter, . Here, we apply a Gaussian approximation to the posterior density using the Laplace method [37], [104], [106], [107]: The filter mean, , is identified with a mode of the posterior density as , and the filter covariance, , is determined from the Hessian of the log posterior at the mode as . The approximate posterior mode is obtained using the iterative procedure of a gradient ascent method or the Newton-Raphson method using the gradient and Hessian matrix. The gradient and Hessian of the log of the posterior density at is calculated as(28)(29)Note that, as in Eqs. 17 and 19, the first derivative of the log normalization function, , with respect to provides the dual coordinates, : . Furthermore, the second derivative yields the Fisher metric: . In this study, we adopt the Newton-Raphson method:(30)The gradient and Hessian, Eqs.28 and 29, are evaluated using an old value, . Here, is a learning coefficient that was introduced in the context of ‘natural’ gradient search algorithm [91]. For the small population size analyzed in this study, we used . However, it is recommended that a smaller positive value be selected for the analysis of a larger system to avoid numerical instability. Because both the likelihood function and prior density are logarithmically concave functions, the posterior is also log-concave. Thus, this optimization problem is convex, which guarantees a unique solution for the filter estimate [90]. The optimized natural parameter is selected as the filter mean, . The filter covariance is approximated as by using . It is also possible to use a simple gradient ascent method to obtain the mode, and then compute the Hessian matrix at the mode.

The above method is equivalent to solving the following nonlinear recursive filter formulas:(31)(32)Eq. 31 was obtained from . Eqs. 31 and 32 are recursively computed in combination with the one-step prediction equations, Eqs. 25 and 26, for . Because and are dual representations of the same probability distribution, Eq. 31 is a nonlinear equation and needs to be solved as suggested above (Eqs. 28, 29, 30). As pointed out for a point process adaptive filter [37], [69], the residual, , in Eq. 31 acts similarly to an innovation vector of a standard Kalman filter. The same error signal between the observed synchrony rates and expected synchrony rates is utilized in training the Boltzmann machine [46], [47], [91]. The innovation term corrects the one-step prediction mean, , i.e., an expected state by the prior distribution. The degree of the correction is determined by the number of repeated trials, , and uncertainty of the predicted state, . The hyper-parameters of the prior density significantly affect the latter gain (see Eq. 26), and thereby the smoothness of the estimated processes. The filter covariance equation, Eq. 32, describes the reduction of the prediction uncertainty, , by observing the parallel spike sequences at time , with the amount determined by the number of repeated trials, , and the Fisher information, . Please see Figure 1 for a geometric view of the recursive Bayesian filter.

Finally, we compute the smooth posterior density using the backward recursive algorithm [68], [105], [107],(33)Because the density functions in the recursive formula were approximated as a normal distribution, we follow the fixed-interval smoothing algorithm [37], [68], [105] established for a Gaussian state and observation equation. Starting from and , which are obtained from the filtering algorithm, we obtain the smoothed mean and covariance,(34)(35)with(36)for . The lag-one covariance smoother, , which appears in the -function, is obtained using the method of De Jong and Mackinnon [108]:(37)

M-step: Optimization of hyper-parameters.

At the M-step, we optimize hyper-parameter given the posterior density under the principle of maximizing the -function. From , the update rule of the covariance matrix, , is obtained as(38)Here, , , and are the smoother mean and covariance, and the lag-one covariance matrix given by Eqs. 34, 35, and 37, respectively. Similarly, from , the auto-regressive parameter, , is updated according to(39)The mean of the initial distribution is updated with from . The covariance matrix, , is not updated; instead, we use a fixed matrix, . Alternatively, the covariance matrix of the initial distribution can be updated according to from , while the mean, , is fixed. Both methods work well with appropriate choices for the fixed parameters. In this study, we updated the mean vector, , of the initial normal distribution, and used a fixed diagonal matrix for its covariance matrix, . It was also suggested to use a stationary mean and covariance of an unconstrained state process (Eq. 8) as parameters of the initial prior distribution [68], [109]. The equilibrium condition from Eq. 8 yields and . The solutions are obtained as and (p. 121 and p. 426 in [109], p. 112 in [110]). In the latter, denotes the Kronecker product (tensor product) and the operator creates a single column vector from a matrix by stacking its column vectors. However, the stationary condition is not always satisfied. As we found that the fitted sometimes violates the stationary assumption, we did not adopt this approach in the current study.

Analysis of Bayesian model comparison method using simulated stationary spike sequences.

In this subsection, we analyze the Bayesian model comparison method by using simulated stationary spike sequences. Figure 8 illustrates an analysis where the BF is applied to stationary spike sequences of simulated neurons. To demonstrate the kind of correlation scenarios that is distinguished by the BF, we generate stationary spike sequences using log-linear models of different spike correlation structures in three distinct time periods (I–III, Figure 8A). These are: (I) an independent model ( for and for at ), (II) a model containing simultaneously positive pairwise interaction terms, without a triple-wise interaction ( for , for , and at ), and (III) a model containing a positive triple-wise interaction, with negative pairwise interactions ( for , for , at ). A sample of the spike sequences from a single trial () is shown in Figure 8B. In all of the models, the first order log-linear parameters, (), are adjusted such that the spike rates of the individual neurons are (). Therefore, the time segments cannot be distinguished on the basis of the firing rates. In addition, the model in period III is designed such that the projection to yields zero pairwise interactions, i.e., the projection yields an independent model. In this model, the second-order joint synchrony rates, (), are at the chance level expected from the individual spike rates, (). However, the model expresses excess joint spike synchrony between all three neurons, , which is larger than the expected rate (0.001). The presence of a positive triple-wise correlation, combined with the absence of marginal pairwise correlations, leads to a very sparse number of excess synchronous spike triplets, as illustrated by the dot displays in period III in Figure 8B. In contrast, a period containing only positive pairwise correlations without a triple-wise interaction (see Figure 8B, period II) contains a relatively large number of triplets, which are, however, the mere expression of excess pairwise correlations.

We calculate the BFs to test for the presence of the two different correlation structures, i.e., the same tests discussed in the Results section: Test 1 checks for the presence of simultaneously positive pairwise interactions and Test 2 checks for the presence of a triple-wise spike correlation. In one test (Test 1) we ask whether the three neurons simultaneously exhibit positive pairwise interactions by applying a model that contains up to pairwise interactions () to each of the three time periods. In that case, the BF (Eq. 48) is computed using a model which assumes that all of the pairwise terms are positive: , specified as for all , as opposed to a model where at least one of the pairwise interactions is not positive: , specified as for at least one . Time (bin) index, , expresses that the BF is computed under the same models for all time steps in the respective time period. In both models, we do not make specific assumptions for the first order log-linear parameters; thus, they are allowed to be real numbers. In the other test (Test 2), we ask whether the neurons exhibit excess triple-wise synchrony by applying a full model () to each of the periods. Here, the BF is computed with the opposing models: , specified as , and , specified as . The lower-order log-linear parameters have again real values.

Figure 8C displays the average BFs (in units of bits) computed from 200 samples (each sample contains trial) within each of the periods I–III. In addition, we show the standard deviations of the BFs (shown as ) derived from the BFs of the 200 realizations. The upper panel shows the result of Test 1, expressing the weight of evidence for the presence of excess synchrony realized by simultaneously positive pairwise interactions. We find substantial evidence for the simultaneously positive pairwise interactions in period II, but not in the other two periods. The lower panel displays the evidence for the presence of a positive triple-wise correlation as evaluated by Test 2. The existing positive triple-wise interaction present in period III is well detected by the large BF, and the very low BFs in period I and II correctly detect the absence of triple-wise correlation.

In Figure 8D, we examine how much each individual spike pattern contributes to the calculation of the BFs discussed above. The weight of evidence (log of the BF) in an observation period is obtained by summing the pieces of evidence computed locally at time using Eq. 52, as . Here we elucidate the contributions of each individual spike pattern to the BF by sorting the bin-by-bin evidence, , with each spike pattern. Figure 8D displays the average BFs for each spike pattern in the three periods in bit unit. The degree of the contribution by a specific spike pattern to the evidence for the tested spike interaction model varies depending on the context in which the spike pattern is observed. For example, the magnitudes of the BFs for a triple-wise correlation (Test 2) found by observing a spike triplet () vary greatly across the three periods (see dark gray bars at in periods I, II, and III). In period III, where pair-synchronous spikes are rarely observed because of the existence of negative pairwise interaction terms, the observation of triplet spikes provides substantial evidence of a triple-wise correlation. In contrast, in period II, where pair-synchronous spikes frequently occur because of the existence of positive pairwise interactions, the chance triplet spikes do not provide substantial evidence of a triple-wise correlation. These results come from the fact that an unexpected synchronous spike pattern that significantly alters and updates the filter density from its one-step prediction density gives rise to a large absolute BF value (Eq. 52). By the same logic, the BFs for the and patterns (and and , not shown) are small because they do not substantially change the posterior densities. However, they should not be neglected because these patterns are abundant as a result of the low firing rates. For example, in period I in Figure 8C, the accumulation of the small weight of evidence from the abundant spike patterns (such as and ) offsets the large weight of evidence induced by a few chance coincidences such as triplet spikes (), thus producing virtually zero weight of evidence for this period.