Broad-band Oscillations by Probabilistic Cellular Automata

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Aug 6, 2008 - where ai,j(t) is an influence to node i from node j at t with possible values. 0 and 1. An example of the influences of neighborhood ai or Λi, could be. Λi = {ai,i−1 ..... Tutorial. Int. J. Bifurc. Chaos, 2:451. [16] W. J. Freeman. (1999).
Broad-band Oscillations by Probabilistic Cellular Automata M ARKO P ULJIC1?, ROBERT KOZMA1 Department of Computer Science, University of Memphis, USA August 6, 2008

Electroencephalographic and magnetoencephalographic potential generated by active brain reveals widespread coherent oscillations. The oscillations of multiple rhythms overlap in broad spectrum noise. Spectrum analysis of short segments reveals peaks in various frequency ranges. As the duration of segments chosen for analysis increases, the basic form to which spectra converge is the one of Brown noise. Any frequency can be modeled with probabilistic cellular automata. Coupled inhibitory and excitatory probabilistic cellular automata with appropriate topology and probabilistic rule governing the local interactions simulate an oscillator. Coupled oscillators create a model with the activation whose spectrum of long segments converges to a Brown noise. Key words: oscillations, PCA, coupled oscillators, chaos

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INTRODUCTION

As the connections among biological neurons grow, the neurons start to group themselves in the semi-isolated populations communicating via neural connections [11]. Single neuron activations seem asynchronous, but the activations of neural assemblies exhibit synchrony [14, 15, 34, 33]. Ordinary and partial differential equations used for neural population modeling show complex dynamics, including chaos and chaotic itinerancy [2, 21, 30]. With the ?

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brain development, some excitatory or stimulating neurons transform into inhibitory neurons, and as such suppress the activity of neurons they influence. Inhibitory neurons form populations as well. The coupling of inhibitory and excitatory populations leads to the emergence of sustained narrow-band oscillations in the neural tissue [4, 5]. Electroencephalographic (EEG) and magnetoencephalographic (MEG) potentials generated by active brain reveal oscillations overlapping with multiple rhythms [16]. Spectrum analysis of short segments shows peaks in the frequency ranges of the θ (3-7 Hz), α (8-12 Hz), β (13-30), and γ (30-100 Hz) bands. If the duration of segments chosen for analysis increases, the form to which the spectrum converges is a linear decrease in log power with increasing log frequency at a slope near 2 (Brown noise). Any frequency range can be modeled with the probabilistic cellular automata (PCA) composed of the inhibitory (reversely influential) and excitatory sites and local and global (long-range) interactions. PCA are lattice models of spatially extended systems with probabilistic local dynamical rules of evolution [18] and they generalize deterministic cellular automata (DCA) and bootstrap percolations [3, 12, 1, 31, 26, 9]. PCA is difficult to analyze rigorously [6], so the computational simulations provide an alternative tool [8, 7, 27]. The role of non-local interactions has been studied in deterministic models as well, such as coupled map lattices (CML) [32, 22, 17]. Stimulation of coupled inhibitory and excitatory populations (oscillator) disturbs the population’s activation amplitudes. Without the stimuli, the population’s activations return to the basal levels. Coupled oscillators of different rhythms create an activation whose spectrum of long segments converges to a desired broad band with the scale-free properties, in the style of brains.

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COUPLED PCA

Node Description

Neuron’s output is the frequency of action potentials generated by inputs, which are the pulses received at the synapses. In essence, the input-output relationship is a non-linear and modeled by a node. A node, cell, or site is the extreme simplification of a neuron with initial activation 0 or 1. The activation of the i-th node at time t+1, ai (t+1) for t = 0, 1, 2, ... and i = 0, 1, 2, ..., N , depends on 2 factors; influences or links of its neighborhood or the nodes with 2

a direct connection to it at time t, Λi , and 0 < ε < 1:   0 with chance 1 − ε; 1 with chance ε   X  | Λi |    if ai,j (t) <   2   ai,j ∈Λi               1 with chance 1 − ε; 0 with chance ε X | Λi | ai (t + 1) = if ai,j (t) >   2   ai,j ∈Λi             0 or 1 with chance 0.5    X  | Λi |    if ai,j (t) =   2

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ai,j ∈Λi

where ai,j (t) is an influence to node i from node j at t with possible values 0 and 1. An example of the influences of neighborhood ai or Λi , could be Λi = {ai,i−1 , ai,i , ai,i+1 }. 2.2 Layer Couplings Population of neurons that are more densely connected are modeled with the layers. A layer is a special 2-dimensional lattice with N nodes. A layer’s topology depends on the nodes’ neighbors. With local topology a node has 5 local neighbors connected by local links; 4 closest in each directions and itself. With non-local topology, some nodes have randomly assigned nonlocal or remote neighbors connected via remote links. To keep the number of neighbors per node constant, a node loses as many local neighbors as the remote neighbors it gains (Fig. 1). To create a periodic boundary, the layer is folded into a torus. The relative importance of non-local links is defined as nR = {# of sites with a non-local link}/{# all sites} ×100%. In a single layer, with neighborhoods of 5, nodes influence with 1 when active and 0 when inactive. Update rule is applied simultaneously over all sites at discrete time steps, so equation 1 is reduced to the probabilistic majority rule, according to which the i-th site next value ai (t + 1) is equal to the majority of its neighbors with probability 1 − ε, and to the minority with probability ε. The layers are coupled if the nodes from one layer neighbor or have cross links to the nodes of another [29]. The higher the number of nodes neighboring the other layer’s nodes, the higher the layer’s coupling strength. One 3

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OSCILLATOR v v v v g g w w a node from layer 0 x x x x sends 0 if 0 else 1 y y yB y B B v v Bv v w w g g a node from layer 1 x x x x sends 1 if 0 else 0 y  y  y y

COUPLED OSCILLATORS oscillator 0 oscillator 1 v v v v v v v v g g w w gB gB w w Q x x x x Q x Bx Bx x layer 0 Q y y yB y y yy y Q B  Q  B Q  QQv v v v  v v Bv v w w g g w w g g    x x x x x x x x layer 1  i  y y  y  y y y i FIGURE 1 Middle left: Illustration of local and non-local links over a square lattice of size 4 × 4; nR = 25% and d = 25%. Top right: PCA composed of two layers (lay.) with nL = 12.5%. Layer 0, (upper) has local topology, so its nR = 0 and nLEI = nLIE = 12.5%. Layer 1 has 4 nodes with remote connections, so its nR = 25%. Bottom: 2 coupled oscillators (osc.). For whole system nO = 6.25%. All connections are bi-directional. E.g. layer 0 of left oscillator, or oscillator 0, is described by its parameters: nR = 0%, nLEI = 12.5%, and nOEI,0,1 = 6.25%.

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of the layers is inhibitory or inversely influential. A node from the inhibitory layer influences the other layer’s node in a reversed manner, influencing with 1 when inactive, and with 0 when active. The other layer’s nodes are excitatory, influence with 1 when active and with 0 when inactive (Fig. 1). A constant size of neighborhood is maintained, so the node loses a connection (its self-connection) when it is selected to receive connection from the other layer. General parameter describing the layers’s coupling strength nL = {# of sites with a cross-layer link}/{# all sites} ×100%. More precisely, nLEI defines amount of layer connections from inhibitory layer to the excitatory. Similarly, nLIE defines amount of layer connections from excitatory layer to the inhibitory. Since coupled excitatory-inhibitory layers can produce oscillation, they are called oscillators. The oscillators are coupled if the nodes from one oscillator neighbor or have cross links to the nodes of another (Fig. 1). The higher the number of nodes neighboring the other oscillator’s nodes, the higher the oscillator’s coupling strength. General parameter describing the oscillator’s coupling strength nO = {# of sites with a cross-oscillator link}/{# all sites} ×100%. Cross-oscillator link’s influence depends on the type of the layer it projects from. More precisely, nOEI,i,j defines amount of layer connections from inhibitory layer of oscillator j to the excitatory layer of oscillator i. Similarly, nOEE,i,j defines amount of layer connections from excitatory layer of oscillator j to the excitatory layer of oscillator i. 3

ACTIVATION BEHAVIOR OF SINGLE AND COUPLED LAYERS

3.1 Single Layer PN System’s activation, d = i ai /N , changes with the  change and the change of other parameters. Single layer PCA with various update rules and neighborhoods have been studied extensively using renormalization group techniques [7, 10, 27] and Binder’s method [8]. Mixed local and non-local or global neighborhoods show critical behavior and follow finite-size scaling laws. Accordingly, magnetization d∗ ∝ (εC − ε)β , when ε → εC . Here d∗ = |d − 0.5|. Similar scaling laws are valid for susceptibility χ and correlation length ξ. The critical exponents satisfy a hyper-scaling identity, which indicates a behavior consistent with weak Ising universality class [28, 24]. Example of the size invariance of quantity U ∗ is illustrated on Fig. 2 in the case of a purely local system. Quantity U ∗ is related to the kurtosis and it is defined as follows: h|d − hdi|4 i (2) U∗ = h(d∗ − hd∗ i)2 i2 5

At the critical point εC , U ∗ is the same for all layer sizes of the same topology [27]. This is the point where the activation densities randomly hover around 0.5 and activation density distributions are uni-modal. For ε < εC , the activations are either mostly active or mostly inactive and the activation distributions are bimodal, as illustrated on the gray shaded histogram on Fig. 2. 3.2 Oscillator The behavior of 2-layer perturbed PCA produce damped oscillation (Fig. 3), which is a basic building block of the dynamics of neural populations [14]. After an impulse stimulus, the excitatory cells reach a peak activation. Then the excitatory cells excite inhibitory cells, which reach a peak excitation a quarter of cycle after the excitatory cells peak activity. At this time, the excitatory cells are already inhibited to their basal, resting level. They reach maximum inhibition as the inhibitory cells return to their basal level. During this phase, the inhibitory cells fail to receive the excitation from the excitatory cells, so they undergo inhibition. When the excitatory cells are released from inhibition, they again respond to background activity, and start another cycle. Inhibitory connections can generate narrow-band oscillations. There are two critical points εO and εC . εO marks the onset of prominent narrow-band oscillations, and εC describes the transition point where narrow-band oscillations disappear. To achieve the statistical accuracy, experiments are performed on lattice sizes between 64 × 64 and 128 × 128 for various ε, nR and nL , and at least for 1 million steps or until h|0.5 − d|i < 0.001. Figures 4 and 5 illustrate the onset of prominent oscillations in systems with nR = 25%, while nL = 3.125% and 12.5%. nLEI = nLIE and nR of the layers are the same. Figures show behavior of one layer, layer 0. The other layer’s behavior is the same. At ε = 0.1425 and nL = 12.5% (dotted line) PCA has a bimodal distribution and produces prominent narrow-band oscillations. At the same time, at nL = 3.125% (shade) PCA has a more complex quadro-modal distribution without narrow-band oscillations. The two plots show U ∗ values for system size 80 × 80 (crosses) and 112 × 112 (squares). The curves intersect the first time when activation density distributions transform from quadromodal to bimodal distribution. This point is marked by (εO ). For given nR , εO decreases as nL increases. For ε < εO , larger coupled layers have greater U ∗ . At εO , the U ∗ values are the same for any system size. Smaller systems have greater U ∗ , immediately above εO . By increasing ε > εO or nL , the frequency of the oscillations increases and the distance between the peaks of the activation distribution decreases, (Fig. 6). When ε approaches εC , the bi6

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FIGURE 7 By increasing nL the frequency of oscillations increases; activation distribution (top) and examples of activations and power spectral density estimates on window of 10,000 steps (bottom) and 1 million step experiment.

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B fiUBN A - fout = G(v)  K0   X v= wi fi

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I 6 ? Pyramidal Cells FIGURE 8 Left: K0 sets defined by a linear 2nd order differential equation and a non-linear sigmoid transformation G(v), which represents wave density. The inputs fi are weighted by weights wi that can be positive (excitatory; E) or negative (inhibitor; I). Sum of the inputs, v, represents pulse density. A K0 set may have both excitatory and inhibitory inputs, however all outputs from a single K0 set act as one type only. Left-down: K2 set as the interaction of two K1 sets, each of which is a layer of coupled K0 sets. The upper layer consists of pure excitatory units and the lower layer consists of inhibitory units. K0 sets within a layer are fully connected. Cross-layer links exists between corresponding nodes. Right: An example of schematic view of the multilayer structure of K3 representing olfactory system. There are excitatory-inhibitory pairs of layers in the Olfactory Bulb (Mitral layer - ML and Granule layers - GL), the Anterior Olfactory Nucleus (AON), and the Prepyriform Cortex (PC).

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4.2

Multiple PCA Oscillators

PCA models have ability to replicate the dynamics of K-sets and thus replace ODE as a modeling tool. Figure 9 illustrates PCA ability to model olfactory system. Three types of weakly coupled oscillators influence each other with their basal frequencies creating the unpredictable mix. In another example, we could couple 8 oscillators with 2 layers each, one excitatory and one inhibitory, (Fig. 10). For each layer nR = 25% and nL = 25%. nOIE,i,j is 1.3125%, 2.625%, 3.9375%, 5.25%, 6.5625%, 7.875%, 9.1875%, and 10.5% for j = 0, 1, 2, 3, 4, 5, 6, 7 respectively, (j 6= i). In other words, inhibitory layers of lower frequency oscillators have more incoming connections from higher frequency oscillators. To create different oscillations, the ε values are 0.0753125, 0.0859375, 0.0965625, 0.1071875, 0.1178125, 0.1284375, 0.1390625, and 0.1496875 for oscillators 0 through 7 respectively. When the oscillators are isolated without the oscillatory connections, each oscillator has its basal frequency, (Fig. 10). When the oscillators are not too strongly coupled, they influence each other, but not enough to give up their own frequency. Almost all layers show a 1/f α overall behavior, which could be indication of chaotic behavior.

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CONCLUSION

Rigorous mathematical analysis of PCA is difficult [6], so PCA behavior is analyzed by computational simulations. Excitatory nodes in a single layer show aperiodic activity, which is self-stabilized by a non-zero point attractor. This activity gives rise to a field of nearly white noise. Coupled inhibitory and excitatory layers with appropriate topology and ε show narrow-band oscillations. The oscillators can model any frequency range. Through U ∗ and statistical analysis, critical points εO and εC are nicely identified. εO and εC mark the onset and the disappearance of prominent narrow-band oscillations, respectively. Coupled oscillators of different frequencies show activations whose spectra converge to broad-band oscillations with scale-free properties. PCA models have the ability to mimic the dynamics of K3 sets, which can demonstrate deterministic chaos [20, 19]. An example in which the oscillators are weakly coupled, so they influence each other, but not overly to lose they own frequencies, show a 1/f α behavior with −3 < α < −2. Coupled PCA described are building blocks which eanble design of the mechanisms which compute in a brain-like fashion. This empowers the future machines. 15

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FIGURE 9 Oscillator 0 parameters: nR = 25% for both layers, nLEI = nLIE = 3.125%, nOIE,0,1 = 3.125%, nOIE,0,2 = 3.125%, and ε = 0.156. Oscillator 1 parameters: nR = 100% for both layers, nLEI = nLIE = 25%, nOEE,1,0 = 3.125%, nOIE,1,2 = 3.125%, and ε = 0.171. Oscillator 2 parameters: nR = 25% for both layers, nLEI = nLIE = 25%, nOEE,2,0 = 3.125%, nOEE,1,1 = 3.125%, and ε = 0.141. Power spectral density averaged on 1 million iterations and window size of 10,000.

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FIGURE 10 Activation behavior of 8 coupled oscillators. nR = 25% and nL = 25%. nOIE,i,j is 1.3125%, 2.625%, 3.9375%, 5.25%, 6.5625%, 7.875%, 9.1875%, and 10.5% for j = 0, 1, 2, 3, 4, 5, 6, 7 respectively, (j 6= i). ε values are 0.0753125, 0.0859375, 0.0965625, 0.1071875, 0.1178125, 0.1284375, 0.1390625, and 0.1496875 for oscillators 0 through 7 respectively. Up: Comparison of few non-coupled and coupled oscillators. Down: Most common power spectrum of layers in coupled oscillators with the diverse ε values.

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