Noise-Mediated Intermittent Synchronization of Collective Behaviors in the Probabilistic Cellular Automata Model of Neural Populations Marko Puljic1 and Robert Kozma1 1 Computational
NeuroDynamics Lab, University of Memphis, Memphis, TN 38152
[email protected],
[email protected]
Abstract Emergence of collective behaviors in populations of interacting units is a central issue of biological self-organization. Recent studies indicate that brains exhibit frequent transitions between states with low and high synchrony extending across the cerebral hemisphere. Such transitions are considered to as hallmarks of high-level cognitive functions and intelligent behavior. Previously, we have developed the neuropercolation model, which is a random cellular automaton with phase transitions induced by noise in a lattice with non-local interactions. In the present work we extend our studies to include interaction of excitatory and inhibitory neural populations. We study in detail the spatio-temporal effects generated by various factors, including noise level, lateral re-wiring topology causing small-world effects, and density of cross-layer connections. We conclude that at a given range of re-wiring and with sparse cross-layer connectivity, the model exhibits resonant type of oscillatory behavior with brief, intermittent de-synchronization of the otherwise highly synchronous oscillations across the whole system. The observed mechanism is a new type of dynamical behavior, which is the property of living substrates operating in a noisy environment with partly non-local interactions. We discuss the implications of these results for interpreting brain experiments and for building artificial machines with intelligent behaviors.
Introduction Synchronization of the firing of widely dispersed neurons in large numbers gives direction to the neural populations that produce behavior (Freeman and Rogers, 2002). In the past years, we have developed the neuropercolation model to describe certain properties of the filamentous tissue called neuropil (Kozma et al., 2004; Balister et al., 2005). Earlier works showed the critical role noise plays in the generation of phase transitions in neuropercolation models (Kozma et al., 2001; Kozma et al., 2002). We have shown that rewiring part of the connections, which leads to small-world effects in the cortical model, can serve as a control mechanism to keep the brain dynamics near criticality (Puljic and Kozma, 2003; Kozma et al., 2005; Puljic and Kozma, 2005). In the neural networks, many neurons send electric potentials to each other and produce the activation levels as observed in EEG experiments (Arhem et al., 2000; Stam et al.,
2003). To measure the synchrony among the neural units, their activation levels are recorded in time and compared. In the present work, first we introduce the neuropercolation model, which includes the interaction of excitatory and inhibitory neural populations. The negative feedback component in such systems is crucial for the generation of complex oscillatory behaviors. We use a twin 2-dimensional lattice arrangement for the excitatory and inhibitory layers. Next we conduct extensive simulations with the model. In particular, we study in detail the spatio-temporal effects generated by various factors, including noise level, lateral re-wiring topology, and density of cross-layer connections. We conclude that at a given range of re-wiring and with sparse cross-layer connectivity, the model exhibits resonant type of oscillatory behavior with brief, intermittent desynchronization of the otherwise highly synchronous oscillations across the whole system. In the discussion section we characterize the observed mechanism, which is a new type of dynamical behavior base don intermittent phase transitions. Such phase transitions characterize systems at the edge of criticality (Bak, 1996; Kaneko and Tsuda, 2001), is the property of living substrates operating in a noisy environment with partly non-local interactions. We conclude with the implications of these results for the interpretation of brain experiments and directions of future research.
Random Cellular Automata Model with Excitatory and Inhibitory Layers Description of Previous Results with Single Layers A cellular automata layer is a 2-dimensional square lattice. Initially, each node has 5 neighbors, 4 in each directions and the 5-th is itself. Based on a random selection rule, some sites are assigned a remote connection. A remote connection is any non-local direct connection coming from remote sites. To keep the total number of connections a site receives constant (5), a site loses a local connection selected randomly, when it is assigned a remote link. We use a periodic boundary condition, i.e., the layer is folded into a torus. Once the topology is fixed, we assign an initial value to each node, active or inactive, 0 or 1. Next we perform an it-
eration, and update the activations, based on a given update rule. Here we use a noisy majority rule, where the noise level is given by ε. In the case of ε = 0, we get the deterministic majority rule, according to which the site’s activation is determined as the majority of its neighbors. In the case of noisy rule, the probability of the majority rule is (1 − ε), while the minority activation would survive with probability ε. Such systems have been extensively studied and the results are given in (Kozma, 2003; Puljic and Kozma, 2003; Puljic and Kozma, 2005; Kozma et al., 2001; Balister et al., 2005), and the results are briefly summarized here. The activation density at time j is defined as the number of active sites divided by the total number of sites. In the experiments, we use square lattices of size up to 512x512. We showed the existence of a critical point, called εc with phase transition. To find the critical regions renormalization group methods (Swendsen, 1979; Swendsen, 1982; Cerf and Cirillo, 1999) and Binder’s methods (Binder, 1981) are exploited. Renormalization explains the effects of system size change on the parameters that describe it. Binder’s methods uses the fact that for the layers with the same structure, (% of remote connections), but different sizes, kurtosis values of activation density as a function of ε and time t differ if ε 6= εc . There is a critical point, εc , where kurtosis value is same for all layer sizes.
Description of Coupled Layers Here the case of two coupled layers is described. The arrangement of the layers is shown in figure 1. Cross-layer connection (lc) is unidirectional, similarly to the lateral connections inside a layer. When there is a layer connection to the site, the site’s self-connection is lost, so the site still has five neighbors. One of the layers is inversely influential (inhibitory). This means that the site from the inhibitory layer affects the other layer in a reversed manner, i.e., influencing with 1 when it is inactive, and with 0 when it is active. The excitatory layer influences the other layer in a usual excitatory way as described in the previous section. For the sake of simplicity, the cross-layer connectivity is symmetrical, i.e., the corresponding (lc) coefficients in the excitatory and inhibitory layers are the same. More details about the oscillations in coupled probabilistic layers can be found in (Puljic et al., 2004).
Experiments with Synchronization in Coupled Layers General Approach Experiments have been conducted with twin excitatoryinhibitory layers of size 256 × 256. At the end of the experiments, the lattice is divided into 16x16 size ‘channels’, and the average density is calculated for each channel. The results given in this section concern the evaluation of the 256 channels, which constitute the lattice. Layers are run
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Figure 1: An example of coupled lattices. A site influenced from another layer loses self-connection, so the neighborhoods are of five.
for ten thousand steps, and at each step the activation level is recorded for the lattice as a whole and for each of the 256 channels. Next we have determined an epoch, which is used in selecting a time window during which the activations of channels and the ensemble are compared. Epoch is a cycle of oscillation if there is one, or a conveniently chosen and time interval otherwise. The time window for the evaluation is chosen typically as 2 epochs. Each channel is compared with the ensemble average at a given time window to evaluate synchrony measures across space and time. Measures of synchrony have been recorded for each channel at each time step. Then the time window was shifted. We have conducted synchrony evaluations both in time domain through correlation functions, and in frequency domain by calculating power spectra. For the sake of compactness of this paper, we do not introduce here results via time domain analysis, but concentrate on the frequency domain. Clearly, the present FFT-based analysis does not give a complete picture and has limitations, especially in the case of highly nonstationary signals near the critical points.
Synchronization Measured via FFT Phase Activations have been generated for ten thousand time steps over lattices of size 256x256; see figure 2 and 3. The center frequency of the dominant ensemble oscillation has been determined using discrete Fast Fourier transform (FFT ). The number of time steps is then divided by the center frequency of the ensemble in order to determine the durations of an epoch, as well as the window size (twice the epoch duration). The ensemble oscillations provide the reference signal. The phases of all channel signals with respect to the ensemble average are determined using the discrete Fourier transform. Fast Fourier transform is calculated for each channel over the given window. Then, the FFT matrix of each channel is multiplied with the FFT matrix of the ensemble over the corresponding segment to correlate the individual channel
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Figure 3: Up: example of channel not synchronous with ensemble (top), and channel synchronous with ensemble (bottom). Down: typical snap shot of phase lags of not synchronous channels with respect to the ensemble. Black; phase lag is −π. White; phase lag is π. and the ensemble. The center frequency of this result is calculated again using FFT. The correlated values are used to locate the phases of channel activations with respect the ensemble average. The results are put into a new matrix, as
Simulations with a wide range of lattice topologies and noise levels have revealed that the coupled excitatory and inhibitory layer model can exhibit prominent oscillations either in a narrow frequency band or in a more broad-band, depending on the experimental parameters. In this section, the observed oscillatory effects are analyzed in details. We have evaluated twin lattices of excitatory and inhibitory units of size 128x128 each. The cross-later coupling coefficient values were selected lc = 12.5%, 25%, and 50%. For each lc value, lateral layer topologies with the following connections have been used: local, 25%(1), and 100%(1). The notations are: 25%(1) means 25% of the sites have exactly one remote connection, while 100%(1) means every site has one remote connection. Accordingly, all together we have conducted 9 sets of experiments. In each set, we fixed the topology, and selected a given noise level. For that noise level we run a simulation and calculated the power spectral density function (PSD) for the time series of the ensemble average activity density in the excitatory lattice. Then we repeated the experiments with another noise level. PSD functions were calculated for a series of 20,000 time points with a window of 2048 points using standard FFT-based signal processing tools. An example of the obtained power spectra are shown in figure 5. The lattice parameters are: local lateral connectivity, and lc = 0.125, 0.25, and 0.50. The noise level is 0.14, which is slightly above the critical value of 0.134 in a single local lattice. We can see an oscillation peak in the spectra in all cases. The overall intensity of oscillations decreases as the lateral connectivity increases. We have evaluated the corresponding standard deviation of the oscillations, which are found to be 533, 423, 274 for lc = 0.125, 0.25, and 0.5, respectively. Please note that the density fluctuations on the 128 x 128 lattice are bounded between 0 and 16384. It is also seen that the frequency of the oscillations drastically
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increases from 1.5 to 10 Hz, as lc increases. Note, that time units are arbitrary in our numerical simulations. For the sake of convenience, we use a value of 1ms for each iteration step, which gives the frequency values as indicated on the figures. Similar analysis has been conducted for lattices with rewiring values 25%(1) and 100%(1). For the sake of brevity, we do not show the corresponding PSDs, but the conclusions are similar to the one derived from figure 5. Figure 6 shows the overall behavior of the power spectral densities (PSD) for various lattice topologies. Each of the 9 plots have been drawn using a sequence of 10 log(PSD) functions calculated with increasing noise levels. The log(PSD) is shown in color (gray) scale, ranging from 103 to 101 0, which appear on the plots as values between 3 to 10 using log scale. The PSDs exhibit some prominent oscillatory peaks. The one exception is the left-most top display on Figure 6, which corresponds to local lattices with sparse cross-layer links (12.5%). The peak becomes sharper and shifts towards higher frequencies as the cross-layer link increases. Next we discuss the range of noise over which oscillations occur. We observe that the oscillatory peaks are bounded both from above and from below. In the case of local lateral connectivity, as well as for the highest cross-layer connectivity lc = 50%, the lower bounds are close to zero. Those are not seem in the displays as they lie out of the analyzed parameter range. Importantly, the lower bound is clearly visible for the lattice with 100%(1) lateral connections and with lc = 0.125, and its neighboring plots. The relatively narrow range of noise level to produce resonance oscillations indicates, that noise can be used as a tuning mechanism to generate and fade away oscillations for the given lattice geometry.
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Conclusions Punctuated Synchronization Synchronization in a biologically motivated cellular automata model are analyzed. Our results show that without remote connections channels cannot synchronize. As the number of remote connections increases, the channels become capable of synchronization. When the noise level ε exceeds a critical level, channels will become desynchronized. In fact, we have identified a range of lattice parameters, for which there is a well-defined noise level with especially favorable conditions of synchronization. For another family of parameter values including ε values, the channels are sometimes in the synchrony and sometimes out of synchrony. In this particular situation, the channels are self-synchronizing and self-de-synchronizing at apparently irregular intervals, following a broad band of oscillatory rhythms, even though the variables describing the system do not change. The self-organizing synchronization and desynchronization sequence is especially significant as it reflects a key property of neural systems with intermittent oscillations. Intermittent phase transitions characterize systems at the edge of criticality, which is the property of living substrates operating in a noisy environment with partly non-local interactions. Brains show such itinerancies, which are related to higher-level cognitive functions. In the future we will extend the present work to build more complex artificial systems, with the potential of intelligent behaviors. Initial results with implementations in robotic platforms are in progress.
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