Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 204

give the same answers to a narrowly specified set of questions. The question we are
primarily interested in here is:
Question 1. Which neurons in the network will fire during which episode along a
given trajectory?
In [ 5 ], we considered a general class of ODE neuronal models and proved rigorous
results for when there is a direct correspondence between the ODE model, M , and a
discrete model, N . More precisely, we demonstrated that dynamic clustering occurs
for all trajectories of M that start in a certain region U of the state space of M and there
is a discrete-time model N that correctly predicts which neurons will fire during each
episode, throughout any ODE trajectory that starts in U . We refer to such a situation
by saying that N is consistent with M on U. Moreover, we may expect the set U ,on
which the two models are consistent, to be open and large enough to contain, for each
possible initial state s ( 0 ) of N ,astateof M that maps to s ( 0 ) .Inthiscasewewillsay

that M realizes N on U. In [ 5 ] we proved the following:
Theorem 1. There exists a class of ODE models M for neuronal networks such that
every model in this class realizes a corresponding discrete model on an open subset
U of its state space and, conversely, every discrete network model N

, 1 = ,

D p

with constant vectors p is realized by some model M in this class.

Figure 6.6 illustrates an example of an ODE model that realizes the network

, 1 , 1 , where D is the digraph of Figure 6.2 b. This network contains seven

cells and the top left panel shows the membrane potentials of each of these cells for
one particular solution. The middle left panel is a grey scale representation of this
solution. Two other solutions are shown in the right panel. In order to generate the
three solutions shown in Figure 6.6 , we chose different initial conditions. Note that
each cell's membrane potential typically lies in one of two “states”; the elevated or
active state corresponds to the firing of an action potential and is represented by the
dark rectangles in the grey scale representations. Moreover, there are discrete episodes
in which some subpopulation of cells lie in the elevated state. These subpopulations
change from episode to episode and the model exhibits dynamic clustering. The dis-
crete model completely predicts which cells fire during which episode for a large
class of initial conditions. The corresponding trajectories in the discrete model are
represented by the active cells in each episode.
In the online supplement, we give a computer code for the ODE model that gen-
erates the solutions shown Figure 6.6 . This code uses the software XPPAUT, which
can be freely downloaded from the webpage [ 18 ]. The code is flexible enough so
that one can explore ODE models that realize different networks. A discussion of
why the model is designed as it is—that is, what the various dependent variables and
parameters correspond to—is well beyond the scope of this chapter. The interested
reader should consult either [ 3 ]or[ 5 ], where this model is described in detail.
As a consequence of Theorem 1 , the results presented in this chapter are
relevant for a general class of neuronal models. An important conclusion of our
study of discrete-time models is that properties of the population rhythm, including

N = D