Influence of Criticality on 1/f α Spectral Characteristics of Cortical Neuron Populations

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1 Influence of Criticality on 1/f α Spectral Characteristics of Cortical Neuron Populations Robert Kozma Computational Neurodynamics Laboratory, Department of Computer Science 373 Dunn Hall, University of Memphis, Memphis, TN Abstract - Critical properties of dynamical models of neural populations are studied. Synchronization of the firing of widely dispersed neurons enables the emergence of spatial patterns of cortical activity. Using neuropercolation model introduced in the literature in the past few years, the dynamics of neural populations is studied near critical regimes. Behavior is evaluated with respect to long-range axonal density and sparseness of feedback between excitatory and inhibitory populations. The results show that 1/f α spectral behavior emerges near criticality, where exponent α is function of the critical state of the system. I. INTRODUCTION Experimental studies indicate that intermittent synchronization across large cortical areas provides the window for the emergence of meaningful cognitive activity in animals and humans (Freeman, 2000; Freeman, 2005; Stam, 2005). Various studies have been aimed at the development of models that interpret the experimental findings. Successful approaches include ordinary differential equations with distributed parameters and partial differential equationsl; see, e.g., Kozma & Freeman (2001, 2003); Steyn-Ross et al. (2005). As an alternative to these classical approaches, in the past years we have developed the neuropercolation model to describe properties of the filamentous cortical tissue called neuropil. Earlier works showed the critical role noise plays in the generation of phase transitions in neuropercolation models (Puljic & Kozma, 2003; Kozma et al., 2004; 2005; Puljic & Kozma, 2005). We have shown that re-wiring part of the connections leads to small-world effects in the cortical model, and it can serve as a control mechanism to keep the brain dynamics near criticality. In neuropercolation models, the synchrony of neural activation is punctuated by episodes with loss of coherence. Neuropercolation models improve understanding of the relationship between the system's stochastic components with homogeneous rules of behavior and the emergent intelligent behavior. In neural tissues, populations of neurons send electric currents to each other and produce activation potentials observed in EEG experiments. The synchrony among neural units can be evaluated by comparing their activation levels as the function of time. While single unit activations have large variability and do not seem synchronous, the activations of neural groups exhibit apparent synchrony. In the case of model calculations, the simulated neural network can be coarse-grained into groups of neurons and the group activation levels tested for synchrony. This is the strategy of the numerical studies followed in the present work. In the present work, we briefly introduce the neuropercolation model and its major features. An emphasis of the present work is on analyzing 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. Spatio-temporal effects and intermittency in this system has been studied by Puljic & Kozma (2006). Here we analyze in details the behavior of the power spectral densities near the critical state with phase transitions in the lattice model. We conclude that 1/f α behavior in power spectra carries sensitive indication on the criticality of the system. This conclusion can be beneficial in the interpretation of EEG and EMG data obtained in human cognitive experiments. II. OVERVIEW OF NEUROPERCOLATION MODEL Here we summarize the basics of neuropercolation approach; for details see, e.g., Puljic & Kozma, 2005, and Kozma et al., We consider a cellular automata layer in a 2-dimensional square lattice. Initially, each node has 5 local neighbors, 4 in each direction (x+, x-, y+, y-) and the 5-th connection is to 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. This is called partial rewiring. To keep the total number of connections a site receives

2 constant, 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 iteration, i.e., we update the activations, based on a given rule. In this work we use a noisy majority rule, where the noise level is given by as a model parameter ε. In the case of zero noise level (ε = 0) the deterministic majority rule states that each site's activation is determined as the majority of its neighbors. In the case of noisy rule with noise level ε, the probability of the majority rule is (1- ε), while the minority activation would survive with probability ε. In our models ε is a small number, typically much less than the unity. We showed the existence of a critical point with phase transition in the mean field limit (Balister et al, 2005). Properties of local model have been thoroughly studied and the presence of phase transition has been proven mathematically for small noise levels (Walters et al, 2005). The case of general neuropercolation model poses a very difficult mathematical problem. Its rigorous mathematical analysis is beyond reach at the present. Computational simulations can provide help to understand the dynamics in the general case. Such systems have been extensively studied and the results are given in Kozma et al. (2005), Puljic & Kozma (2005). An important issue is to find the critical regions and describe the model behavior near criticality. To this aim the renormalization group approach and Binder's method are exploited (Binder, 1981). Renormalization explains the effects of system size change on the characteristic parameters. Binder's methods uses the fact that for the layers with the same structure but different sizes, the kurtosis (4 th statistical momentum) of activation density is invariant of the system size at the state of criticality. Based on this property, critical states on neuropercolation models can be identified and thoroughly characterized; see; e.g., Puljic (2006). III. EXPERIMENTAL CONDITIONS Here the case of two coupled layers is described, one layer is excitatory and the other is inhibitory. Note that all connections are unidirectional, i.e., the ones across layers and inside a given layer as well. When there is a connection is established to a given site from the other layer, the site's self-connection is lost. Therefore, the site still has five neighbors. One of the layers is inhibitory. This means that the site from the inhibitory layer affects the other (excitatory) 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 pattern is symmetric, and the corresponding coupling coefficients in the excitatory and inhibitory layers are the same. Experiments have been conducted using a twin layer of squared lattices of various sizes, up to 256 x 256. The model has several control parameters: Noise level (ε): characterizing the randomness of the update rule, i.e., noisy majority voting; Lateral rewiring probability (γ): which is the relative proportion of non-local connections describing long-range axonal effects; Cross-layer connection probability (η): the density (sparseness) of excitatory and inhibitory links. It is important to note that any of the above parameters can be used to control phase transitions. Fixing two of the parameters, a phase transition can be induced by changing the remaining parameter. For example, for zero rewiring (γ = 0) and zero cross-layer connection density (η = 0), we obtain the single layer local random cellular automaton with the noise (ε) as control parameter. The activation density at time t is defined as the number of active sites divided by the total number of sites. We conduct experiments with a given number of iteration steps. In order to calculate the critical parameters with proper accuracy, exhaustive experiments are required, which include often millions of iterations. The present work, however, is directed toward the qualitative characterization of the oscillations using power spectral densities near the critical point. We consider cases far below critical (sub-critical), near critical, approximately critical, and above critical (super-critical). For the purposes of the present analysis, we conduct experiments with up to 10^5 iterations. The noise level ε has been varied from several percentages up to 21%. Rewiring has been conducted from γ = zero (local model) to 25%. Connection density η between excitatory and inhibitory nodes has been varied up to 50%. Previous studies show that excessive excitatory-inhibitory connections yield very strong limit cycle oscillations (Puljic & Kozma, 2006). Our goal is to study biologically realistic, relatively broad-band oscillations, therefore, we avoid high values of cross-layer connection density. The results introduced in this work correspond to just a few % in terms of η. The present results indicate that interesting and potentially biologically interpretable results can be obtained with the given parameter choice. IV. RESULTS ON 1/f α SPECTRAL BEHAVIOR Experiments with η = 12.5% cross-layer connection density and γ = 6.25% rewiring rate are shown in Figure 1. The first column shows examples of the produced time series, while the second column includes the normalized power spectral densities (NAPSD).

3 Figure 1. Lattice dynamics; rewiring rate is 6.25 %; Left column time plots of lattice activation density, right column normalized power spectral densities. Noise level from top to bottom: 12%, 15%, 17%, and 18%, corresponding to subcritical, near but below critical, critical, and supercritical regimes.

4 Figure 1. Lattice dynamics; rewiring rate is 25.0 %; Left column time plots of lattice activation density, right column normalized power spectral densities. Noise level from top to bottom: 16%, 19%, 20%, and 21%, corresponding to subcritical, near/below critical, critical, and supercritical regimes.

5 NAPSDs have been calculated using the activation density, defined as the ratio of the active sites and the total number of sites. The maximum density is 1 when all the sites are active and zero when all are inactive. The power spectral densities are calculated using Fourier transformed time series with a window of 2048 points. Based on evaluating the open-loop transfer functions, the iteration time step has been taken to be 0.05 ms. The noise level has been selected as follows in Figure 1, starting from top row towards the bottom: 12%, 15%, 17%, and 18%. These particular values have been selected as 12% corresponds to deeply subcritical state; 15% is a state below critical but close to criticality. 17% is very near criticality, although we did not attempt to fine tune system parameters to reach criticality. Finally, 18% is a state in the supercritical regime. We can see that the activation density is close to white-noise with a near-flat spectrum over the studied frequency range up to 250 Hz. As we approach criticality, strong oscillations emerge with sharp resonance peak at around 40 Hz. Further increasing the noise level, the oscillation becomes more broad-band. It is remarkable that at (or very close to) the critical regime the PSD exhibits 1/f α behavior over the gamma and high gamma band, with a slope in the range of -2 to -3. In the case of supercritical noise, the PSD become very flat with some decay at higher frequencies. Similar conclusion can be drawn from Fig. 2 where the case of γ = 25% is shown. Note that the general conclusions are the same as in Fig.1, but the oscillations in the gamma band tend to shift towards higher frequencies. It can be concluded that the rewiring level is an important control parameter to influence high-frequency oscillations and the critical slope of the power spectra. V. CONCLUDING REMARKS We have analyzed the 1/f α behavior of power spectra of neural populations modeled by the neuropercolation approach. We studied the role of 3 control parameters, namely the noise level σ, rewiring rate γ, and density η of feedback connections between excitatory and inhibitory layers. Critical combinations of the parameters can produce conditions of phase transitions in the model. The spectra are rather flat in the subcritical and supercritical regimes. However, they exhibit clear 1/f α behavior near criticality. The exponent α depends on the actual critical parameters. Previous studies (Puljic & Kozma, 2006) indicate that without remote connections the channels cannot synchronize. As the number of remote connections increases the channels become capable of synchronization. There are noise levels, for which the channels are sometimes in the synchrony and sometimes out of synchrony, even though the variables describing the system do not change. Intermittent synchronization among cortical areas has been identified as hallmarks of higherlevel cognitive behavior. The present study indicates that by monitoring the slope of the power spectra an important measure of the criticality and related synchronization effects can be established. Acknowledgments: This work was supported in part by grant NSF EIA VI. REFERENCES P. Balister, B. Bollobas, R. Kozma (2005) Mean Field Models of Probabilistic Cellular Automata, Random Structures and Algorithms (in press). K. Binder (1981) Z. Phys. B, 43. Freeman, W.J. (2000) Neurodynamics An exploration of mesoscopic brain dynamics. Springer Verlag, London. Freeman, W.J. (2005) Biological Cybernetics (in press). Kozma, R. & W.J., Freeman (2001) Chaotic resonance: Methods and applications for robust classification of noisy and variable patterns. Int. J. Bifurcation & Chaos, 11(6): Kozma, R., W.J. Freeman, (2003). Basic principles of the KIV model and its application to the navigation problem, Journal of Integrative Neuroscience, 2, Kozma, R., Puljic, M., Balister, P., Bollob as, B., and Freeman, W. J. (2004). Neuropercolation: A random cellular automata approach to spatio-temporal neurodynamics. Lecture Notes in Computer Science, 3305: Kozma, R., Puljic, M., Balister, P., Bollob as, B., and Freeman, W. J. (2005). Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions. Biological Cybernetics, 92(6): Puljic, M. and Kozma, R. (2003). Phase transitions in a probabilistic cellular neural network model having local and remote connections. In International Joint Conference on Neural Networks IJCNN, pages Puljic, M. and Kozma, R. (2005). Activation clustering in neural and social networks. Complexity, 10(4): Puljic, M. (2006) Neuropercolation Models for Dynamics of Interacting Populations, PhD Dissertation, University of Memphis. Puljic, M., Kozma, R. (2006) Noise mediate Intermittent Synchronization of behaviors in the Random Cellular Automaton Model of Neural Populations, ALIFEX, June 2-6, 2006, Bloomington, IN, MIT Press (in press) Steyn-Ross, D.A., Steyn-Ross, M.L., Sleigh, J.W., Wilson, M.T., Gillies, I.P., Wright, J.J. The Sleep Cycle Modelled as a Cortical Phase Transition, Journal of Biological Physics 31: , 2005.

6 Stam, C.J., (2005) Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field, Clinical Neurophysiology 116 (2005) Walters, M., Balister, P., Bollobas, B. (2005) Majority Percolation, (submitted)

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