An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized `locked' solutions is analyzed.

We explore a computationally efficient method of simulating realistic networks of neurons introduced by Knight, Manin, and Sirovich (1996) in which integrate-and-fire neurons are grouped into large populations of similar neurons. For each population, we form a probability density which represents the distribution of neurons over all possible states. The populations are coupled via stochastic synapses in which the conductance of a neuron is modulated according to the firing rates of its presynaptic populations. The evolution equation for each of these probability densities is a partial differential-integral equation which we solve numerically. Results obtained for several example networks are tested against conventional computations for groups of individual neurons. We apply this approach to modeling orientation tuning in the visual cortex. Our population density model is based on the recurrent feedback model of a hypercolumn in cat visual cortex of Somers et al. (1995). We simulate the response to oriented flashed bars. As in the Somers model, a weak orientation bias provided by feed-forward lateral geniculate input is transformed by intracortical circuitry into sharper orientation tuning which is independent of stimulus contrast. The population density approach appears to be a viable method for simulating large neural networks. Its computational efficiency overcomes some of the restrictions imposed by computation time in individual

Present and permanent address: Physik-Department der TU Munchen Exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire. If the postsynaptic potential is decreasing, oscillations are bound to be unstable. This is a kind of locking theorem and boils down to a subtle interplay of axonal delays, postsynaptic potentials, and refractory behavior. The theorem also allows for mixtures of excitatory and inhibitory interactions. On the basis of the locking theorem we present a simple geometric method to verify existence and local stability of a coherent oscillation. 2 1

The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons. We study synchronization in models of large networks of spiking neurons with random sparse connectivity. Synchrony occurs only when the average number of synapses, M , that a cell receives is larger than a critical value, M c . Below M c , the system is in an asynchronous state. In the limit of weak coupling, assuming identical neurons, we reduce the model to a system of phase oscillators which are coupled via an effective interaction, \Gamma. In this framework, we develop an approximate theory for sparse networks of identical neurons to estimate M c analytically from the Fourier coefficients of \Gamma. Our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it. We apply this theory to compute M c for a model of inhibitory networks of integrate-and-fire (I&F) neurons as a function of the intrinsic neuronal properties (e.g., the refractory period T r ), the synaptic time constants and the strength of the external stimulus, I ext . The number M c is found to be non-monotonous with the strength of I ext . For T r = 0, we estimate the minimum value of M c over all the parameters of the model to be 363:8. Above M c , the neurons tend to fire in: 1) smeared one cluster states at high firing rates and 2) smeared two or more cluster states at low firing rates. Refractoriness decreases M c at intermediate and high firing rates. These results are compared against numerical simulations. We show numerically that systems with different sizes, N , behave in the same way provided the connectivity, M , is such a way that 1=M eff = 1=...

A class of synchronization protocols for dense, large-scale sensor networks is presented. The protocols build on the recent work of Hong, Cheow, and Scaglione [5, 6] in which the synchronization update rules are modeled by a system of pulse-coupled oscillators. In the present work, we define a class of models that converge to a synchronized state based on the local communication topology of the sensor network only, thereby lifting the all-to-all communication requirement implicit in [5, 6]. Under some rather mild assumptions of the connectivity of the network over time, these protocols still converge to a synchronized state when the communication topology is time varying. Categories and Subject Descriptors

Introduction In this chapter we will discuss some practical and technical aspects of numerical methods that can be used to solve the equations that neuronal modelers frequently encounter. We will consider numerical methods for ordinary differential equations (ODEs) and for partial differential equations (PDEs) through examples. A typical case where ODEs arise in neuronal modeling is when one uses a single lumped-soma compartmental model to describe a neuron. Arguably the most famous PDE system in neuronal modeling is the phenomenological model of the squid giant axon due to Hodgkin and Huxley. The difference between ODEs and PDEs is that ODEs are equations in which the rate of change of an unknown function of a single variable is prescribed, usually the derivative with respect to time. In contrast, PDEs involve the rates of change of the solution with respect to two or more independent variables, such as time and space. The numerical methods we will discuss for both ODEs and

A novel approach to cortical modeling was introduced by Knight et al. (1996). In their presentation cortical dynamics is formulated in terms of in- teracting populations of neurons, a perspective that is in part motivated by modern cortical imaging (For a review see Sirovich and Kaplan (2002)). The approach

A Globally Coupled Map (GCM) model is a network of chaotic elements that are globally coupled with each other. In this paper, first, a modified GCM model called the "Globally Coupled Map using the Symmetric map (S-GCM)" is proposed. The SGCM is designed for information-processing applications. The S-GCM has attractors called "cluster frozen attractors," each of which is taken to represent information. This paper also describes the following characteristics of the S-GCM which are important to information-processing applications: (a) The S-GCM falls into one of the cluster frozen attractors over a wide range of parameters. This means that the information representation is stable over parameters; (b) Represented information can be preserved or broken by controlling parameters; (c) The cluster partitioning is restricted, i.e., the representation of information has a limitation. Finally, our techniques for applying the S-GCM to information processing are shown, considering these characteris...

this article, I will consider only a gaussian distribution of membrane depolarizations of magnitude p.h/ p 2 .h N h/ 2 2 2 (2.13) or the nonstochastic limit 0, in which case h/; (2.14) where .x/ is the Dirac distribution. I will refer to

Introduction Spiking neurons are highly non-linear oscillators. As such they display collective behavior that may have important calculational manifestations. Synchronization between the firing of different neurons is the first topic to which we devote our attention. This behavior can be brought about in our integrate-and-fire model through excitatory synaptic couplings without delays, or inhibitory couplings with delays. Once the mechanism of synchronization is established, this phenomenon can be used for defining data clustering. The clusters correspond to neurons that fire synchronously, with different clusters firing at different times. This behavior can also be described as temporal segmentation, separating data through phase lags between excitations of different aggregates. This separation is characteristically limited to a small number of segments, a limitation that is inherent to the behavior of coupled non-linear oscillators. The importance of synchrony as s