Stimulus space topology vs. network topography in the ring model

Abstract

Event Abstract Back to Event Stimulus space topology vs. network topography in the ring model The allowed patterns of activity among neurons in a recurrent network are constrained by both the structure of inputs and the structure of recurrent connections. Nevertheless, there is a growing body of evidence suggesting that, in the mature brain, patterns of spontaneous activity are similar to patterns of evoked activity, even in the absence of structured inputs. It is therefore possible that one outcome of learning is that recurrent networks serve to constrain activity patterns to be “sensible” - i.e., to reflect the same structure that is normally present during evoked activity even when the inputs are unstructured. If so, what can be inferred about connectivity in recurrent networks whose constraints reflect relations between single-cell receptive fields? We address this question in a simple class of models, in which the function of a recurrent network is to gate inputs so that only a selected set of persistent activity patterns is allowed. By “persistent activity pattern” we mean a subset of stably co-active neurons. An elegant feature of these models is that one can analytically determine the set of all stable steady states (“permitted sets”) from the synaptic matrix alone, and these activity patterns are highly constrained even when the allowed inputs are not [1]. If the allowed activity patterns are consistent with overlapping receptive fields, one can infer topological features of the underlying stimulus space [2]. This allows us to directly relate recurrent network connectivity to the topology of the represented stimulus space. We use this paradigm in an analysis of the ring model [3] for the case of unconstrained inputs. In this model, neurons are labeled by angles varying along a circle, and the strengths of recurrent connections depend only on the differences in angles between pairs of cells. If the stimulus space topology obtained from neural activity patterns simply reflected the topographic organization of the recurrent network, one would expect a circle topology for the ring model. We find instead that, depending on parameters, there are three possibilities: the topology may be either that of a point, a circle, or too complex to reflect a low-dimensional stimulus space. Only in the case of circle topology are the activity patterns consistent with the usual interpretation that individual cells have convex and overlapping receptive fields, each spanning a limited and continuous interval of angles on a circular stimulus space; this parameter regime determines preferred patterns of connectivity required for the ring model to represent a circular variable in the case of unconstrained inputs. We suggest that knowledge of the stimulus space represented by a recurrent network can provide new insights into network connectivity, even in cases where the topographic organization of the network is known. This work was supported by the Swartz Foundation and NSF DMS-0818227