Characterising neural representation in terms of the dynamics of cell membrane potential activity: a control theoretic approach using differential geometry

Experimental and theoretical results seem to demand that the study of neural representations in the brain considers both the subthreshold and suprathreshold dynamic activity of the neural membrane potential, rather than be solely focussed on stimulus representation in trains of action potentials. In a dynamical systems formulation, the membrane potential can be regarded as the “state” of the neuron, evolving continuously over time and space, within an infinite dimensional space, in response to ever changing inputs. Formally, the state of the neuron, together with future inputs, is sufficient to fully determine the future behaviour of the neuron. In this paper, the characterisation of membrane potential activity is approached from a control theoretic viewpoint as a “reachability” problem, in which the effect of particular stimulus-evoked synaptic inputs is seen as driving the cell from some initial state of the neuron to a particular terminal state on a given manifold. It is shown that a fluctuating subthreshold membrane potential induced by synaptic background activity, and the cooperative interaction of excitatory and inhibitory inputs, may be important factors in allowing the cell to “reach” a maximal subset of all possible membrane potential states, through the action of its synaptic inputs.