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NJIT Mathematical Biology Seminar

Tuesday, February 3, 2009, 4:00pm
Cullimore Hall 611
New Jersey Institute of Technology

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Stimulus space representations by recurrent networks: the geometry of fixed points

Vladimir Itskov

Center for Theoretical Neuroscience, Columbia University

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. 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 model, 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 this model 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. If the allowed activity patterns are consistent with overlapping receptive fields, one can infer topological features of the underlying stimulus space. 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 for the case of unconstrained inputs. 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. The knowledge of the stimulus space represented by a recurrent network thus can provide new insights into network connectivity, even in cases where the topographic organization of the network is known. This is joint work with Carina Curto.