Hebbian theory proposes that ensembles of neurons form a basis for neural processing. It is possible to gain insight into the activity patterns of these neural ensembles through a binary analysis, regarding neurons as either active or inactive. The framework of permitted and forbidden sets, introduced by Hahnloser, Seung, and Slotine (2003), is a mathematical model of such a binary analysis: groups of coactive neurons can be permitted or forbidden depending on the network's structure. In order to widen the applicability of the framework of permitted sets, we extend the permitted set analysis from the original threshold-linear regime. Specifically, we generalize permitted sets to firing rate models in which Φ is a nonnegative continuous piecewise C1 activation function. In our framework, the focus is shifted from a neuron's firing rate to its responsiveness to inputs; if a neuron's firing rate is sufficiently sensitive to changes in its input, we say that the neuron is responsive. The algorithm for categorizing a neuron as responsive depends on thresholds that a user can select arbitrarily and that are independent of the dynamics. Given a synaptic weight matrix W, we say that a set of neurons is permitted if it is possible to find a stimulus where those neurons, and no others, remain responsive. The main coding property we establish about PΦ(W), the collection of all permitted sets of the network, is that PΦ(W) is a convex code when W is almost rank one. This means that PΦ(W) in the low-rank regime can be realized as a neural code resulting from the pattern of overlaps of receptive fields that are convex.

We present a method to distinguish direct connections between two neu- rons from common input originating from other, unmeasured neurons. The distinction is computed from the spike times of the two neurons in response to a white noise stimulus. Although the method is based on a highly idealized linear-nonlinear approximation of neural response, we demonstrate via simulation that the approach can work with a more re- alistic, integrate-and-fire neuron model. We propose that the approach exemplified by this analysis may yield viable tools for reconstructing stimulus-driven neural networks from data gathered in neurophysiology experiments.

A previously developed method for efficiently simulating complex networks of integrate-and-fire neurons was specialized to the case in which the neurons have fast unitary postsynaptic conductances. However, inhibitory synaptic conductances are often slower than excitatory ones for cortical neurons, and this difference can have a profound effect on network dynamics that cannot be captured with neurons that have only fast synapses. We thus extend the model to include slow inhibitory synapses. In this model, neurons are grouped into large populations of similar neurons. For each population, we calculate the evolution of a probability density function (PDF), which describes the distribution of neurons over state-space. The population firing rate is given by the flux of probability across the threshold voltage for firing an action potential. In the case of fast synaptic conductances, the PDF was one-dimensional, as the state of a neuron was completely determined by its transmembrane voltage. An exact extension to slow inhibitory synapses increases the dimension of the PDF to two or three, as the state of a neuron now includes the state of its inhibitory synaptic conductance. However, by assuming that the expected value of a neuron's inhibitory conductance is independent of its voltage, we derive a reduction to a one-dimensional PDF and avoid increasing the computational complexity of the problem. We demonstrate that although this assumption is not strictly valid, the results of the reduced model are surprisingly accurate.

The complexity of neural networks of the brain makes studying these networks through computer simulation challenging. Conventional methods, where one models thousands of individual neurons, can take enormous amounts of computer time even for models of small cortical areas. An alternative is the population density method in which neurons are grouped into large populations and one tracks the distribution of neurons over state space for each population. We discuss the method in general and illustrate the technique for integrate-and-fire neurons.