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Information processing in the brain is mediated by the dynamics of large, highly interconnected neuronal populations. The activity patterns exhibited by the brain are extremely rich; they include stochastic weakly correlated local firing, synchronized oscillations and bursts, as well as propagating waves of activity. My research is aimed at discovering the dynamic mechanisms underlying these spatio-temporal patterns and their role in the neuronal codes used by the brain to communicate and transform signals. My approach is to develop analytical tools and theoretical understanding of the cooperative dynamical states in network models that mimic aspects of the dynamics and architecture of local neuronal populations in the brain. These theoretical studies are based on statistical mechanics, the theory of stochastic systems, and the physics of non-linear dynamical systems. I then extend the theoretical understanding by numerical simulations of more biologically realistic network models. Advancing the understanding of brain dynamics and function can be achieved only by a close interaction between theory and experiment. An essential part of my theoretical investigation is the elucidation of the relevant anatomical and physiological constraints and the derivation of experimentally testable predictions.

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With regards to the first issue my work has concentrated on understanding the role of cooperative network effects in generating rhythmic activity. Unlike in most systems studied in physics the interaction between neurons is not instantaneous. So the analysis of these systems gives us a chance to determine a delay in the interaction affects the network's behavior. We have analyzed the stability of the asynchronous state in large networks of all-to-all coupled non-linear oscillators with pulse coupling [1]. The analysis was done by determining the temporal evolution of the distribution of the neuronal phases, and constructing an eigenvalue equation for small perturbations away from homogeneous distribution that characterizes the asynchronous state. This analysis is exact for arbitrarily strong coupling. Using these methods networks of integrate-and-fire neurons were studied extensively and the nature of the bifurcations from the asynchronous state established [2]. The fully synchronized and its stability was also analyzed. An important conclusion of this work was that, because of the finite rise time of the synaptic inputs, networks of neurons with excitatory coupling never completely synchronize, whereas networks with inhibitory coupling do. For excitatory networks the asynchronous state can be stable, while for networks with inhibitory coupling the asynchronous state is never stable. This is exactly the opposite of what happens in models with instantaneous coupling. Hansel {\it et al.} [3] showed in analytical studies for weak coupling and numerical simulations for both weak and strong coupling the same qualitative behavior is observed in networks of more realistic Hodgkin-Huxley neurons.

An important restriction of the method that was used to analyze the asynchronous state, is that it could only be applied to oscillators that are described by a first order differential equation of a single parameter. Recently I have developed the techniques to extend this analysis to all-to-all coupled networks of non-linear oscillators described by arbitrarily many parameters. An eigenvalue equation, similar to the one in networks of one-dimensional oscillators, determines the stability of the asynchronous state. This eigenvalue equation can be determined from studying how a single neuron that receives constant input reacts to a small transient input.

In this respect the method is similar to the method that is used to study networks of non-linear oscillators with weak coupling, the phase-coupled model [4]. In this model the interaction between the neurons can also be inferred from the single cell characteristics. Unlike in the strong coupling case this method can be used to study any state of the network and is not restricted to the asynchronous state.

We have employed the technique for strong coupling to study networks of integrate and fire neurons with spike adaptation [5,6]. In this work we show that a network of regular spiking neurons with adaptation can exhibit a bifurcation to a state with synchronized burst activity for sufficiently strong coupling. Assuming a large time constant for the adaptation this burst state can also be studied. Since this bursting state is only present in networks with a strong coupling between the cells it cannot be studied in the weak coupling limit. Indeed in the phase-coupling model such a bursting state does not exist. This shows that if the coupling strength is strong, qualitatively different states can arise, and thus methods that go beyond the weak coupling limit are important for the understanding of networks of coupled neurons.

Using techniques form statistical dynamics of disordered systems, we have shown that sparsely connected networks with relatively strong synapses and deterministic dynamics can evolve to a state in which the cells fire highly irregularly, even with a constant, non-fluctuating input [7,8]. The reason that the cells in such a network fire irregularly, is that the network goes to a state in which the total excitatory input is roughly canceled by the total inhibitory input. Because the synapses are relatively strong, both the total excitatory and the total inhibitory currents are large compared to the threshold. Both have fluctuations that are small compared to there mean, but because they are nearly balanced, the fluctuations are not small compared to the average net input. Thus the fluctuations can bring the input above the threshold at irregular intervals. In this respect the network behaves very similar to extremely diluted spin-glasses [9]. However unlike in spin-glasses, where the balance is assured by the randomness in the sign of the connections, in this model the balance is a dynamical property of the network, that is reached because the network evolves to an operating point in which the excitatory and inhibitory rates are such that the mean excitatory and inhibitory inputs nearly cancel. The techniques that were used to study the balanced state allowed us to determine several characteristics of this state. We have shown that this state very naturally leads to a distribution of rates that is skewed, with some cells firing at a significantly higher rates. Such rate distributions are commonly observed in cortical ensembles, though up to now nobody had a good explanation why rate distributions have this shape. An important functional advantage of the balanced state is that the network can follow changes in external input very rapidly.

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Another feature that should be relaxed is the extreme dilution we have assumed up to now. The fraction of the cells that project to each cell in the network should agree with the anatomical and electrophysiological data. If the the probability of synaptic connections is increased, the balanced model come closer to the model of Hansel and Sompolinsky [11,12], that investigated synchronized irregular activity. This model relies on a high degree of synchrony rather than strong coupling to achieve non-negligible fluctuations in the cells inputs. This opens up the possibility of having irregular firing through a mechanism that lies between the two scenarios of strong coupling with very weak synchrony and weaker coupling with very strong synchrony. I intend to explore the connection between these two scenarios.

To address these issues close collaboration with electrophysiologists is necessary, since many of parameters that are necessary for more realistic models are at present not known.

while it is unclear that the balance of inhibition and excitation is the mechanism that generates the high variability of the firing of cortical neurons, one expects that the mechanism that generates high variability in the cortex, will result in correlated firing between neurons in a cortical column and the neurons that project to this network, particularly when the activity of these neurons is correlated. Assuming that the input neurons that code ' for a single stimulus behave in a correlated manner, one can study how this correlation affect the behavior of the cells in the column, by investigating the response of balanced networks that receive correlated input. Thus the balanced network model mig ht provide an excellent tool to develop methods to study coding in the cortex. In particular it will provide a useful model to assess the meaning of cross-correlations commonly measured experimentally.

In the models that have been studied analytically the activity of individual cells is very regular if the network activity is periodic. In models with strong coupling only all to all coupling has been considered. This results in all cells getting the same inputs, leading to regular activity of the cells. In the phase coupled model one can consider networks with random dilution in the coupling, but since in this model the coupling is very weak, the cells fire periodically. While our combined theoretical understanding of all to all coupled networks with strong coupling and randomly coupled networks with phase coupled interaction will give some insight in oscillation in cortical networks , what is really needed is a model with coupling that is both strong and randomly diluted.

I plan to study such networks analytically, using a combination of techniques from dynamical system theory and statistical mechanics of disordered systems. I also want to develop methods to study states other than the asynchronous state in networks with strong synaptic coupling to understand the cortical properties of networks that are not firing asynchronously.

As the models become increasing more realistic one will be less and less capable to determine their properties analytically. Thus we will have to rely more on numerical simulations. But even for the interpretation of numerical simulations it is very useful to have good understanding of simpler systems that can be studied analytically.

If we have theories that can describe functional networks with irregularly firing neurons as well as a theory that describes networks of irregularly firing networks with oscillatory network activity, we can determine how going to an oscillatory state affects the functional role of the network. So even though the research I propose to do is mainly concerned with the mechanisms that underly the observed activity in the cortex, the work could also have strong implications for the functional aspect of cortical activity.

One of the issues I want to study in this system is the 40 Hertz oscillations 14]. These oscillations do most likely result from interactions with different cortical areas and the thalamus. This raises the question how rhythmic activity can arise from the cooperation of different pools of neurons. While these oscillations have been studied extensively, no satisfactory explanation of them has offered. It is my hope that with a better theoretical understanding of rhythmogenesis in simple model systems we will be able to resolve this issue better. A striking property of these oscillations is that they disappear when the subject pays attention to the visual scene. This leads one to suspect that these oscillations actually interfere with perception. But if this is the case, why are they there in the first place?

It is a well known, but little understood, fact that cells that are far apart from each other in the primary visual cortex can show synchronized activity when they are responding to parts of a continuous contour in the visual field [15]. The long range interactions between different hypercolumns is probably important for establishing this synchrony. It is commonly believed that this synchrony helps later stages in the visual system to bind line segments that belong to one object. As such this synchrony is extremely important for the functioning of the visual system. Could it be that the wiring that makes this synchrony possible leads to the 40 Hertz oscillations when attention is not focused on the visual input? If this is the case, it would have strong implications in general for the cortex, since 40 Hertz oscillations are observed in many cortical areas.

To address these questions close cooperation with experimentalists is essential. By far to many of the parameters that would go into models that can address these issues are at the moment unknown, and without these theoretical models it would probably not even be clear that these properties might be important. Also the predictions that would come out of these models have to be tested experimentally to test the validity of these models.

[1] Abbott, L.F. and van Vreeswijk, C.A.
** Phys. Rev. E**48:1483 (1993).

[2] van Vreeswijk, C.A.
** Phys. Rev. E**54:5522 (1996).

[3] Hansel, D., Mato, G., and Meunier, C.
** Europhys. Lett.** 23:367 (1993); ** Phys. Rev. E**48:3470
(1993); ** Neural Comp.** 7:307 (1995).

[4] Kuramoto, Y. * Chemical Oscillations, Waves and
Turbulence*, (Springer, New York, 1984).

[5] Goldberg, J., Hansel D., and van Vreeswijk C.
** Israel J. of Med. Sci.** 32:S22. (1996).

[6] van Vreeswijk, C and Hansel D. (1998) in preparation.

[7] van Vreeswijk, C.A. and Sompolinsky, H.
** Science** 274:1724. (1996).

[8] van Vreeswijk, C.A. and Sompolinsky, H
** Neural Computation** 10:1321. (1998).

[9] Derrida, B., Gardner, E., and Zippelius, A.
** Europhys. Lett.** 4:167. (1987).

[10] Gilbert, G., and Wiesel, H.
** J. Neurosci.** 3:1116. (1983).

[11] Hansel, D and Sompolinsky, H
** Phys. Rev. Lett.** 68, 718. (1992).

[12] Hansel, D and Sompolinsky, H
** J. Comput. Neurosci.** 3, 7. (1996).

[13] Holt, G.R., Softky, W.R., Koch, C., and
Douglas, R.J. ** J. Neurophysiol.** 75, 1806. (1996).

[14] Gray, C.M. and Singer, W.
**Nature** 338,334. (1989).

[15] Gray, C.M. ** J. Comp. Neurosc.** 1, 11.
(1994) and references therein.

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