Wilson–Cowan model

About: Wilson–Cowan model is a research topic. Over the lifetime, 34 publications have been published within this topic receiving 5153 citations.

A Mathematical Theory of the Functional Dynamics of Cortical and Thalamic Nervous Tissue

Abstract: It is proposed that distinct anatomical regions of cerebral cortex and of thalamic nuclei are functionally two-dimensional. On this view, the third (radial) dimension of cortical and thalamic structures is associated with a redundancy of circuits and functions so that reliable signal processing obtains in the presence of noisy or ambiguous stimuli. A mathematical model of simple cortical and thalamic nervous tissue is consequently developed, comprising two types of neurons (excitatory and inhibitory), homogeneously distributed in planar sheets, and interacting by way of recurrent lateral connexions. Following a discussion of certain anatomical and physiological restrictions on such interactions, numerical solutions of the relevant non-linear integro-differential equations are obtained. The results fall conveniently into three categories, each of which is postulated to correspond to a distinct type of tissue: sensory neo-cortex, archior prefrontal cortex, and thalamus. The different categories of solution are referred to as dynamical modes. The mode appropriate to thalamus involves a variety of non-linear oscillatory phenomena. That appropriate to archior prefrontal cortex is defined by the existence of spatially inhomogeneous stable steady states which retain contour information about prior stimuli. Finally, the mode appropriate to sensory neo-cortex involves active transient responses. It is shown that this particular mode reproduces some of the phenomenology of visual psychophysics, including spatial modulation transfer function determinations, certain metacontrast effects, and the spatial hysteresis phenomenon found in stereopsis.

The Wilson–Cowan model, 36 years later

Abstract: The Wilson–Cowan model of interacting neurons (1973) is one of the most influential papers published in Biological Cybernetics (Kybernetik). This paper and a companion paper published in 1972 have been cited over 1000 times. Rather than focus on the microscopic properties of neurons, Wilson and Cowan analyzed the collective properties of large numbers of neurons using methods from statistical mechanics, based on the mean-field approach. New experimental techniques to measure neuronal activity at the level of large populations are now available to test these models, including optical recording of brain activity with intrinsic signals and voltage sensitive dyes, and new methods for analyzing EEG and MEG. These measurement techniques have revealed patterns of coherent activity that span centimetres of tissue in the cerebral cortex. Here the underlying ideas are reviewed in a historic context.

Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics

Abstract: We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit N -> infinity, where N determines the size of each population, the dynamics is described by deterministic Wilson–Cowan equations. On the other hand, for finite N the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady–state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite N, we calculate the exponentially small rate of noise–induced transitions between the resulting metastable states using a Wentzel–Kramers–Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory/inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles).

A showcase of torus canards in neuronal bursters

Abstract: Rapid action potential generation - spiking - and alternating intervals of spiking and quiescence - bursting - are two dynamic patterns commonly observed in neuronal activity. In computational models of neuronal systems, the transition from spiking to bursting often exhibits complex bifurcation structure. One type of transition involves the torus canard, which we show arises in a broad array of well-known computational neuronal models with three different classes of bursting dynamics: sub-Hopf/fold cycle bursting, circle/fold cycle bursting, and fold/fold cycle bursting. The essential features that these models share are multiple time scales leading naturally to decomposition into slow and fast systems, a saddle-node of periodic orbits in the fast system, and a torus bifurcation in the full system. We show that the transition from spiking to bursting in each model system is given by an explosion of torus canards. Based on these examples, as well as on emerging theory, we propose that torus canards are a common dynamic phenomenon separating the regimes of spiking and bursting activity.