9.2 Dynamic patterns of neuronal activity

Up to now we have treated only a single sheet of neurons that were all of the
same type. Excitatory and inhibitory couplings were lumped together in a
single function *w* that gave the `average' coupling strength of two
neurons as a function of their distance. `Real' neurons, however, are either
excitatory *or* inhibitory, because they can use only one type of
neurotransmitter (Dale's law). A coupling function that yields both positive
and negative values for the synaptic couplings is therefore not realistic.

We can easily extend the previous model so as to account for different
types of neuron or for several separate layers of neuronal tissue. To
this end we embellish the variable *u* for the average membrane
potential with an additional index *k*,
*k* = 1,..., *n*, that denotes
the type of the neuron or its layer. Furthermore, we introduce
coupling functions
*w*_{kl}(*x*, *x'*) that describe the coupling
strength of a neuron from layer *l* at position *x'* to a neuron
located in layer *k* and position *x*. In analogy to Eq. (9.4) the field equations will be defined as

with

w_{11}0, w_{21}0, w_{12}0, andw_{22}0 . |
(9.36) |

For the sake of simplicity we assume that all coupling functions are bell-shaped, e.g.,

with mean coupling strength and spatial extension .

9.2.1 Oscillations

As before we start our analysis of the field equations by looking for
homogeneous solutions. Substitution of
*u*_{k}(*x*, *t*) = *u*_{k}(*t*) into Eq. (9.36) yields

with = d

We can gain an intuitive understanding of the underlying mechanism by means of
phase-plane analysis - a tool which we have already encountered in
Chapter 3. Figure 9.8 shows the flow-field and null-clines of
Eq. (9.39) with = 1, = 5,
= = 2,
= - 1, and
= 0. The gain function has a standard form, i.e.,
*g*(*u*) = {1 + exp[(*u* - )]}^{-1} with = 5 and = 1.

For zero external input Eq. (9.39) has only a single
stable fixed point close to
(*u*_{1}, *u*_{2}) = (0, 0). This fixed point is attractive
so that the system will return immediately to its resting position after a
small perturbation; cf. Fig. 9.8A. If, for example, the external input
to the excitatory layer is gradually increased, the behavior of the systems
may change rather dramatically. Figure 9.8B shows that for
*I*^{ext, 1} = 0.3 the system does not return immediately to its resting
state after an initial perturbation but takes a large detour through phase
space. In doing so, the activity of the network transiently increases
before it finally settles down again at its resting point; cf. Fig. 9.8B. This behavior is qualitatively similar to the triggering of an action
potential in a two-dimensional neuron model (cf. Chapter 3),
though the interpretation in the present case is different. We will refer to
this state of the network as an *excitable* state.

If the strength of the input is further increased the system undergoes a series of bifurcations so that the attractive (0, 0)-fixed point will finally be replaced by an unstable fixed point near (1, 1) which is surrounded by a stable limit cycle; cf. Fig. 9.8C. This corresponds to an oscillatory state where excitatory and inhibitory neurons get activated alternatingly. Provided that the homogenous solution is stable with respect to inhomogeneous perturbations global network oscillations can be observed; cf. Fig. 9.9.

9.2.2 Traveling waves

Traveling waves are a well-known phenomenon and occur in a broad
class of different systems that have collectively been termed *excitable media*. A large class of examples for these systems is
provided by reaction-diffusion systems where the interplay of a
chemical reaction with the diffusion of its reactants results in an
often surprisingly rich variety of dynamical behavior. All these
systems share a common property, namely `excitability'. In the absence
of an external input the behavior of the system is characterized by a
stable fixed point, its resting state. Additional input, however, can
evoke a spike-like rise in the activation of the system. Due to
lateral interactions within the system such a pulse of activity can
propagate through the medium without changing its form and thus
forming a traveling wave.

In the previous section we have seen that the present system consisting of two separate layers of excitatory and inhibitory neurons can indeed exhibit an excitable state; cf. Fig. 9.8B. It is thus natural to look for a special solution of the field equations (9.36) in the form of a traveling wave. To this end we make an ansatz,

u_{k}(x, t) = (x - v t) , |
(9.39) |

with an up to now unknown function that describes the

This is a nonlinear integro-differential equation for the form of the traveling wave. In order to obtain a uniquely determined solution we have to specify appropriate boundary conditions. Neurons cannot `feel' each other over a distance larger than the length scale of the coupling function. The average membrane potential far away from the center of the traveling wave will therefore remain at the low-activity fixed point , i.e.,

with

0 = - + g[] + I^{ext, k} . |
(9.42) |

This condition, however, still does not determine the solution uniquely because Eq. (9.41) is invariant with respect to translations. That is to say, with (

Finding a solution of the integro-differential equation () analytically is obviously a hard problem unless a particularly
simple form of the gain function *g* is employed. One possibility is
to use a step function such as

g(u) = |
(9.43) |

with being the threshold of the activation function. In this case we can use the translation invariance and look for solutions of Eq. (9.41) containing a single pulse of activation that exceeds threshold on a certain finite interval. Since

Figure 9.10 shows an example of a traveling wave
in a network with excitatory (layer 1, = 1) and
inhibitory (layer 2, = 5) neurons. The coupling functions are
bell-shaped [cf. Eq. (9.38)] with
= = = 1 and
= = 2,
= - 1, and
= 0, as before. The excitatory neurons receive tonic
input
*I*^{ext, 1} = 0.3 in order to reach the excitable state
(cf. Fig. 9.8B). A short pulse of additional excitatory
input suffices to trigger a pair of pulses of activity that travel in
opposite direction through the medium.

Cambridge University Press, 2002

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