transmembrane potential

Fig. 2. Two different distributions of neural membrane potentials, with the same mean states, will in general have different mean firing rates (arbitrary units)

field formulations discussed above. In situations where higher order moments of the distribution (variance, kurtosis etc) are retained, the possibility exists for deep, but tractable representations of complex neuronal dynamics. Hence interactions between stochastic and deterministic processes, as embodied in (19) can be formally studied.

2 The Geometry of Dynamics: A Mini-Encyclopedia of Terms

In this section, we step back from a consideration of the forms that evolution equations can take and overview the crucial geometrical means of understanding them.

2.1 Basic Dynamical Concepts

As mentioned in the Introduction, nonlinear differential equations can be notoriously intractable with regards to exact analytic solutions. However, a thorough understanding of their dynamics is very often possible by combining analysis and geometry. In this Section, we provide the central defining terms through the exploration of some simple dynamical systems. In interests of brevity we have sought to explain the intuitive meaning of the terms, keeping technical definitions to a minimum. Most of these terms are given more formal definitions in standard dynamical systems textbooks (e.g. Strogatz 1994). Illustrated examples of all terms follow in subsequent sections.

For any study of geometry, we require a space in which to embed our objects of study. For evolution equations, a manifold fulfills this purpose. Put simply, a manifold is a space which can be locally stretched or deformed into a Euclidean space whilst having a variety of global shapes. Hence the local structure sustains the intuitive meaning of terms such as a neighbourhood (a ball of small radius) which are crucial for issues requiring a "distance", well defined for Euclidean space. The global structure of a manifold, on the other hand, can be quite complicated, and may be 'bounded' (like the unit interval) or 'unbounded' (like the Euclidean plane), 'simply connected' (like a sphere) or not simply connected (like a torus). A differentiable manifold has the additional properties required to support differentiation. The planar surface, a torus and a sphere are differentiable manifolds. Although the properties of these spaces may seem trivial, a formal definition of a differentiable manifold must be able to support quite general dynamical systems. For example, the manifold of a partial differential equation has infinite dimension!

A phase space is a differentiable manifold whose axes are spanned by the dynamical variables Z = {Zi, Z2,Z3,...} of an evolution equation. The topology ("shape") of the phase space is chosen to match the properties of these variables. For example, the plane (RxR) is a suitable phase space for a system with two membrane potentials. For a system where the two variables are phases varying between 0 and 2n, the torus (S x S) is preferable because of the periodic nature of the boundaries.

We can think of a point in phase space as the instantaneous state Z(t) of our system. If we substitute this state into our evolution equation, we would get the instantaneous rate of change of the system dZ(t) /dt when in that state. This defines a tangent vector in the phase space, telling us how the system will evolve into its next state Z(t'). This critical step - of linking dynamics to geometry - is captured by the vector field, a directed flow through a phase space which embodies the evolution equation. More technically, a vector field assigns a vector to every point in phase space which is precisely the solution of the evolution equation at that point. Hence these vectors capture both the rate and direction of change of the system. For example, the vector field corresponding to the trivial one-dimensional equation dx/dt = —x is just the set of all vectors of length x pointing towards the origin. The vector field at x = 1 is a vector of length 1 directed towards the origin.

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