9.3 Patterns of spike activity

We have seen that the intricate interplay of excitation and inhibition
in locally coupled neuronal nets can result in the formation of
complex patterns of activity. Neurons have been described by a
graded-response type formalism where the `firing rate' is given as a
function of the `average membrane potential'. This approach is clearly
justified for a qualitative treatment of *slowly* varying
neuronal activity. In the context of spatio-temporal patterns of
neuronal activity, however, a slightly closer look is in order.

In the following we will dismiss the firing rate paradigm and use the
Spike Response Model instead in order to describe neuronal activity in
terms of individual action potentials. We start with a large number of
SRM neurons arranged on a two-dimensional grid. The synaptic coupling
strength *w* of neurons located at and is,
as hitherto, a function of their distance, i.e.,
*w* = *w*(| - |). The response of a neuron to the
firing of one of its presynaptic neurons is described by a response
function and, finally, the afterpotential is given by a
kernel named , as customary. The membrane potential of a neuron
located at is thus

with

Spikes are triggered whenever the membrane potential reaches the firing threshold . This can be expressed in compact form as

S(, t) = [u(, t) - ] . |
(9.45) |

Here, [...]

Figure 9.11 shows the result of a computer simulation of a network consisting of 1000×1000 SRM neurons. The coupling function is mexican-hat shaped so that excitatory connections dominate on small and inhibitory connections on large distances. In a certain parameter regime the network exhibits an excitable behavior; cf. Section 9.2.2. Starting from a random initial configuration, a cloud of short stripes of neuronal activity evolves. These stripes propagate through the net and soon start to form rotating spirals with two, three or four arms. The spirals have slightly different rotation frequencies and in the end only a few large spirals with three arms will survive.

Let us try to gain an analytic understanding of some of the phenomenon
observed in the computer simulations. To this end we suppose that the
coupling function *w* is slowly varying, i.e., that the distance
between two neighboring neurons is small as compared to the
characteristic length scale of *w*. In this case we can replace in
Eq. (9.45) the sum over all presynaptic neurons by an integral
over space. At the same time we drop the indices that label the
neurons on the grid and replace both *h* and *S* by continuous
functions of and *t* that interpolate in a suitable way between
the grid points for which they have been defined originally. This
leads to field equations
that describe the membrane potential
*u*(, *t*)
of neurons located at ,

together with their spike activity

as a function of time

The approach sketched in Sections 9.3.1 and 9.3.2 (Kistler et al., 1998; Bressloff, 1999; Kistler, 2000) is presented for a network of a single population of neurons, but it can also be extended to coupled networks of excitatory and inhibitory neurons (Golomb and Ermentrout, 2001). In addition to the usual fast traveling waves that are found in purely excitatory networks, additional slow and non-continuous `lurching' waves appear in an appropriate parameter regime (Golomb and Ermentrout, 2001; Rinzel et al., 1998).

9.3.1 Traveling fronts and waves (*)

We start our analysis of the field equations (9.47) and (9.48) by looking for a particular solution in form of a plane front of excitation in a two-dimensional network. To this end we make an ansatz for the spike activity

S(x, y, t) = (t - x/v) . |
(9.48) |

This is a plane front that extends from

u(x, y, t) = (t - x/v) + dx' dy' w (t - x'/v) . |
(9.49) |

Up to now the propagation velocity is a free parameter. This parameter can be fixed by exploiting a self-consistency condition that states that the membrane potential along the wave front equals the firing threshold; cf. Eq. (9.48). This condition gives a relation between the firing threshold and the propagation velocity

u(x = v t, y, t) = dx' dy' w (t - x'/v) |
(9.50) |

Note that the afterpotential drops out because each neuron is firing only once. The propagation velocity as a function of the firing threshold is plotted in Fig. 9.12A. Interestingly, there are two branches that correspond to two different velocities at the same threshold. We will see later on that not all velocities correspond to stable solutions.

The simulations show that the dynamics is dominated in large parts of the net by a regular pattern of stripes. These stripes are, apart from the centers of the spirals, formed by an arrangement of approximatively plane fronts. We can use the same ideas as above to look for such a type of solution. We make an ansatz,

S(x, y, t) = t - , |
(9.51) |

that describes a traveling wave, i.e., a periodic arrangement of plane fronts, with wave length traveling in positive

with

u_{front}(x, y, t) = dx' dy' w (t - x'/v) . |
(9.53) |

Using the fact that the membrane potential on each of the
wave fronts equals the firing threshold we find a relation between the
phase velocity and the wave length. This relation can be reformulated
as a *dispersion relation* for the wave number
*k* = 2/
and the frequency
= 2 *v*/. The dispersion relation,
which is shown in Fig. 9.12B for various values of the firing
threshold, fully characterizes the behavior of the wave.

9.3.2 Stability (*)

A single front of excitation that travels through the net triggers a
single action potential in each neuron. In order to investigate the
stability of a traveling front of excitation we introduce the firing
time
*t*() of a neuron located at . The threshold
condition for the triggering of spikes can be read as an implicit
equation for the firing time as a function of space,

In the previous section we have found that

We are aiming at a linear stability analysis in terms of the firing
times (Bressloff, 1999). To this end we consider a
small perturbation
*t*(*x*, *y*) which will be added to the solution of a
plane front of excitation traveling with velocity *v* in positive
*x*-direction, i.e.,

t(x, y) = x/v + t(x, y) . |
(9.55) |

This ansatz will be substituted in Eq. (9.55) and after linearization we end up with a linear integral equation for

Due to the superposition principle we can concentrate on a particular form of the perturbation, e.g., on a single Fourier component such as

with Re() > 0 then the front is unstable with respect to that perturbation.

Figure 9.13 shows the result of a numerical
analysis of the stability equation (9.58). It turns out that
the lower branch of the *v*- curve corresponds to unstable
solutions that are susceptible to two types of perturbation, viz.,
a perturbation with
Im() = 0 and a oscillatory perturbation
with
Im() 0. In addition, fronts with a velocity larger
than a certain critical velocity are unstable because of a form
instability with
Im() = 0 and > 0. Depending on the
actual coupling function *w*, however, there may be a narrow
interval for the propagation velocity where plane fronts are stable;
cf. Fig. 9.13B.

The stability of plane waves can be treated in a similar way as that
of a plane front, we only have to account for the fact that each
neuron is not firing only once but repetitively. We thus use the
following ansatz for the firing times
{*t*_{n}(*x*, *y*)| *n* = 0,±1,±2,...} of a neuron located at (x,y),

t_{n}(x, y) = + t_{n}(x, y) , |
(9.57) |

with

in leading order of

For the sake of simplicity we neglect the contribution of the after
potential in Eq. (9.60), i.e., we assume that
[*n* /*v*] = 0 for *n* > 0. This assumption is justified for
short-lasting afterpotentials and a low firing frequency.

As before, we concentrate on a particular form of the perturbations
*t*_{n}(*x*, *y*), namely
*t*_{n}(*x*, *y*) = exp[*c* (*x* - *n* )] cos( *n*) cos( *y*). This corresponds to a sinusoidal
deformation of the fronts in *y*-direction described by together
with a modulation of their distance given by . If we substitute this
ansatz for the perturbation in Eq. (9.60) we obtain a set of equations
that can be reduced to two linearly independent equations for *c*, ,
and . The complex roots of this system of equations determines the
stability of traveling waves, as it is summarized in Fig. 9.12.

Cambridge University Press, 2002

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