We present a general approximation method for the mathematical analysis of spatially localized steady-state solutions in nonlinear neural field models. These models comprise several layers of excitatory and inhibitory cells. Coupling kernels between and inside layers are assumed to be gaussian shaped. In response to spatially localized (i.e., tuned) inputs, such networks typically reveal stationary localized activity profiles in the different layers. Qualitative properties of these solutions, like response amplitudes and tuning widths, are approximated for a whole class of nonlinear rate functions that obey a power law above some threshold and that are zero below. A special case of these functions is the semilinear function, which is commonly used in neural field models. The method is then applied to models for orientation tuning in cortical simple cells: first, to the one-layer model with "difference of gaussians" connectivity kernel developed by Carandini and Ringach (1997) as an abstraction of the biologically detailed simulations of Somers, Nelson, and Sur (1995); second, to a two-field model comprising excitatory and inhibitory cells in two separate layers. Under certain conditions, both models have the same steady states. Comparing simulations of the field models and results derived from the approximation method, we find that the approximation well predicts the tuning behavior of the full model. Moreover, explicit formulas for approximate amplitudes and tuning widths in response to changing input strength are given and checked numerically. Comparing the network behavior for different nonlinearities, we find that the only rate function (from the class of functions under study) that leads to constant tuning widths and a linear increase of firing rates in response to increasing input is the semilinear function. For other nonlinearities, the qualitative network response depends on whether the model neurons operate in a convex (e.g., x2) or concave (e.g., sqrt(x)) regime of their rate function. In the first case, tuning gradually changes from input driven at low input strength (broad tuning strongly depending on the input and roughly linear amplitudes in response to input strength) to recurrently driven at moderate input strength (sharp tuning, supralinear increase of amplitudes in response to input strength). For concave rate functions, the network reveals stable hysteresis between a state at low firing rates and a tuned state at high rates. This means that the network can "memorize" tuning properties of a previously shown stimulus. Sigmoid rate functions can combine both effects. In contrast to the Carandini-Ringach model, the two-field model further reveals oscillations with typical frequencies in the beta and gamma range, when the excitatory and inhibitory connections are relatively strong. This suggests a rhythmic modulation of tuning properties during cortical oscillations.
['Thomas Wennekers']
Orientation Tuning Properties of Simple Cells in Area V1 Derived from an Approximate Analysis of Nonlinear Neural Field Models