- 8.2.1 Locking in Noise-Free Populations
- 8.2.1.1 Example: Perfect synchrony in noiseless SRM
_{0}neurons - 8.2.1.2 Example: SRM
_{0}neurons with inhibitory coupling - 8.2.1.3 Derivation of the locking theorem (*)

- 8.2.1.1 Example: Perfect synchrony in noiseless SRM
- 8.2.2 Locking in SRM
_{0}Neurons with Noisy Reset (*)

- 8.2.3 Cluster States

8.2 Synchronized Oscillations and Locking

We have seen in the previous section that the state of asynchronous firing can
loose stability towards certain oscillatory modes that are solutions of the
*linearized* population equations. We are now going to investigate
oscillatory modes in more detail and check whether a
large-amplitude oscillation
where all neurons are firing in ``lockstep'' can be a stable solution of the
population equations.

8.2.1 Locking in Noise-Free Populations

We consider a homogeneous population of SRM_{0} or integrate-and-fire neurons
which is nearly perfectly synchronized and fires almost regularly with period
*T*. In order to analyze the existence and stability of a fully locked
synchronous oscillation we approximate the population activity by a sequence
of square pulses *k*,
*k* {0,±1,±2,...}, centered around
*t* = *k* *T*. Each pulse *k* has a certain half-width and amplitude
(2)^{-1} - since all neurons are supposed to fire once in each
pulse. In order to check whether the fully synchronized state is a stable
solution of the population equation (6.75), we assume that the
population has already fired a couple of narrow pulses for *t* < 0 with widths
*T*, *k* 0, and calculate the amplitude and width of
subsequent pulses. If we find that the amplitude of subsequent pulses
increases while their width decreases (i.e.,
= 0), then we conclude that the fully locked state is stable.

To make the above outline more explicit, we use

as a parameterization of the population activity; cf. Fig. 8.4. Here, (.) denotes the Heaviside step function with (

As we will see below, the condition for stable locking of all neurons in the
population can be stated as a condition on the *slope* of the input
potential *h* at the moment of firing. More precisely, if the last population
pulse occurred at about *t* = 0 with amplitude *A*(0) the amplitude of the
population pulse at *t* = *T* increases, if *h'*(*T*) > 0:

If the amplitude of subsequent pulses increases, their width decreases. In other words, we have the following

The Locking Theorem is applicable for large populations that are already close
to the fully synchronized state. A related but *global* locking
argument has been presented by (Mirollo and Strogatz, 1990). The locking argument can be
generalized to heterogeneous networks (Chow, 1998; Gerstner et al., 1993a) and to
electrical coupling (Chow and Kopell, 2000). Synchronization in small networks has been
discussed in, e.g., (Bose et al., 2000; Hansel et al., 1995; Chow, 1998; Ernst et al., 1995; van Vreeswijk, 1996; van Vreeswijk et al., 1994). For
weak coupling, synchronization and locking can be systematically analyzed in
the framework of phase models (Ermentrout and Kopell, 1984; Kopell, 1986; Kuramoto, 1975) or
canonical neuron models
(Izhikevich, 1999; Hansel et al., 1995; Ermentrout, 1996; Ermentrout et al., 2001; Hoppensteadt and Izhikevich, 1997).

Before we derive the locking condition for spiking neuron models, we illustrate the main idea by two examples.

In this example we will show that locking in a population of spiking neurons can be understood by simple geometrical arguments; there is no need to use the abstract mathematical framework of the population equations. It will turn out that the results are - of course - consistent with those derived from the population equation.

We study a homogeneous network of *N* identical neurons which are mutually
coupled with strength
*w*_{ij} = *J*_{0}/*N* where *J*_{0} > 0 is a positive
constant. In other words, the (excitatory) interaction is scaled with one over
*N* so that the total input to a neuron *i* is of order one even if the number
of neurons is large (
*N*). Since we are interested in synchrony we
suppose that all neurons have fired simultaneously at
= 0. When
will the neurons fire again?

Since all neurons are identical we expect that the next firing time will also
be synchronous. Let us calculate the period *T* between one synchronous pulse
and the next. We start from the firing condition of SRM_{0} neurons

where (

since

What happens if synchrony at *t* = 0 was not perfect? Let us assume that one of
the neurons is slightly late compared to the others; Fig. 8.5B. It
will receive the input
*J*_{0} (*t*) from the others, thus the
right-hand side of (8.14) is the same. The left-hand side, however, is
different since the last firing was at instead of zero. The next
firing time is at
*t* = *T* + where is found from

Linearization with respect to and yields:

Thus the neuron which has been late is `pulled back' into the synchronized pulse of the others, if the postsynaptic potential is rising at the moment of firing at

We see from Fig. 8.5B that, in the case of excitatory coupling, stable locking works nicely if the transmission delay is in the range of the firing period, but slightly shorter so that firing occurs during the rise time of the EPSP.

Locking can also occur in networks with purely inhibitory couplings
(van Vreeswijk et al., 1994). In order to get a response at all in such a system, we
need a constant stimulus *I*_{0} or, equivalently, a negative firing threshold
< 0. The stability criterion, however, is equivalent to that of the
previous example.

Figure 8.6 summarizes the stability arguments analogously to
Fig. 8.5. In Fig. 8.6A all neurons have fired synchronously
at *t* = 0 and do so again at *t* = *T* when the inhibitory postsynaptic potential
has decayed so that the threshold condition,

- (T) = J_{0} (t - k T) , |
(8.17) |

is fulfilled. This state is stable if the synaptic contribution to the potential, (

8.2.1.3 Derivation of the locking theorem (*)

We consider a homogeneous populations of SRM neurons that are close to a
periodic state of synchronized activity. We assume that the population
activity in the past consists of a sequence of rectangular pulses as specified
in Eq. (8.11). We determine the period *T* and the sequence of
half-widths of the rectangular pulses in a self-consistent manner.
In order to prove stability, we need to show that the amplitude *A*(*k* *T*)
increases while the halfwidth decreases as a function of *k*. To
do so we start from the noise-free population equation (7.13) that we
recall here for convenience

where

As a first step, we calculate the potential
*h*_{PSP}(*t*|). Given
*h*_{PSP} we can find the period *T* from the threshold condition and
also the derivatives
*h* and
*h* required for
Eq. (7.13). In order to obtain
*h*_{PSP}, we substitute
Eq. (8.11) in
(6.8), assume
*T*, and integrate. To first order in
we obtain

where - is the last firing time of the neuron under consideration. The sum runs over all pulses back in the past. Since (

In the second step we determine the period *T*. To do so, we consider a
neuron in the *center* of the square pulse which has fired its last spike
at = 0. Since we consider noiseless neurons the relative order of
firing of the neurons cannot change. Consistency of the ansatz (8.11)
thus requires that the next spike of this neuron must occur at *t* = *T*, viz. in
the center of the next square pulse. We use = 0 in the threshold
condition for spike firing which yields

If a synchronized solution exists, (8.20) defines its period.

In the population equation (8.18) we need the derivative of
*h*_{PSP},

According to Eq. (8.18), the new value of the activity at time

We now apply Eq. (8.21) to a population of
SRM_{0} neurons.
For SRM_{0} neurons we have
(*x*, *s*) = (*s*), hence
*h* = 0 and
*h*_{PSP}(*t*|) = *h*(*t*) = *J*_{0}(*t* + *k* *T*). For a standard kernel (e.g. an
exponentially decaying function), we have
(*T*) > 0 whatever *T* and thus

which is identical to Eq. (8.12). For integrate-and-fire neurons we could go through an analogous argument to show that Eq. (8.12) holds. The amplitude of the synchronous pulse thus grows only if

The growth of amplitude corresponds to a compression of the width of the
pulse. It can be shown that the `corner neurons' which have fired at time
± fire their next spike at
*T*±
where
= *A*(0)/*A*(*T*).
Thus the square pulse remains normalized
as it should be. By iteration of the argument for *t* = *k* *T* with
*k* = 2, 3, 4,... we see that the sequence
converges to zero and the square pulses approach a Dirac -pulse under
the condition that
*h'*(*T*) = (*k* *T*) > 0. In other words, the
*T*-periodic synchronized solution with *T* given by Eq. (8.20) is stable,
if the input potential *h* at the moment of firing is rising
(Gerstner et al., 1996b).

In order for the sequence of square pulses to be an exact solution of the
population equation, we must require that the factor
in the square brackets of Eq. (8.18)
remains constant over the width of a pulse. The derivatives of
Eq. (8.19), however, do depend on *t*.
As a consequence, the form of the
pulse changes over time as is visible in Fig. 8.7. The activity as
a function of time was obtained by a numerical integration of the population
equation with a square pulse as initial condition for a network of SRM_{0}
neurons coupled via (8.10) with weak inhibitory coupling *J* = - 0.1 and
delay
= 2ms. For this set of parameters *h'* > 0 and locking
is possible.

8.2.2 Locking in SRM

The framework of the population equation allows us also to extend the locking
argument to noisy SRM_{0} neurons. At each cycle, the pulse of synchronous
activity is compressed due to locking if *h'*(*T*) > 0. At the same time it is
smeared out because of noise. To illustrate this idea we consider SRM_{0}
neurons with Gaussian noise in the reset.

In the case of noisy reset, the interval distribution can
be written as
*P*_{I}(*t*|) = d*r* [*t* - - *T*(, *r*)] _{}(*r*);
cf. Eq. (5.68). We insert the
interval distribution into
the population equation
*A*(*t*) = *P*_{I}(*t*|) *A*() d and find

The interspike interval of a neuron with reset parameter

where

We now search for periodic solutions.
As shown below, a limit cycle solution of
Eq. (8.24) consisting of a
sequence of Gaussian pulses exists if the noise amplitude is small
and
(*h'*/) > 0. The width *d* of the activity pulses
in the limit cycle is proportional to the noise level .
A simulation of locking in the presence of
noise is shown in Fig. 8.8. The network of SRM_{0} neurons has
inhibitory connections (*J*_{0} = - 1) and is coupled via the response kernel
(8.10) with a transmission delay of
= 2 ms.
Doubling the noise level leads to activity pulses with twice the
width.

In order to calculate the width of the activity pulses in a locked state, we
look for periodic pulse-type solutions of Eq. (8.24). We
assume that the pulses are Gaussians with width *d* and repeat with period
*T*, viz.,
*A*(*t*) = _{d}(*t* - *k* *T*). The pulse width *d* will be
determined self-consistently from Eq. (8.24). The integral over *r* in
Eq. (8.24) can be performed and yields a Gaussian with width
= [*d*^{2} + ]^{1/2}. Equation (8.24) becomes

where

Let us work out the self-consistency condition and focus on the pulse around
*t* 0. It corresponds to the *k* = 0 term on the left-hand side which
must equal the *k* = - 1 term on the right-hand side of
Eq. (8.25). We
assume that the pulse width is small *d* *T* and expand *T*_{b}(*t*) to linear
order around
*T*_{b}(0) = *T*. This yields

The expansion is valid if

The Gaussian on the left-hand side of (8.27) must have the same width as the Gaussian on the right-hand side. The condition is

where

8.2.3 Cluster States

We have seen that, on the one hand, the state of asynchronous firing is typically unstable for low levels of noise. On the other hand, the fully locked state may be unstable as well if transmission delay and length of the refractory period do not allow spikes to be triggered during the rising phase of the input potential. The natural question is thus: What does the network activity look like if both the asynchronous and the fully locked state are unstable?

Figure 8.9A shows an example of an excitatory network with vanishing transmission delay and a rather long refractory period as compared to the rising phase of the postsynaptic potential. As a consequence, the threshold condition is met when the postsynaptic potential has already passed its maximum. The fully locked state is thus unstable. This, however, does not mean that the network will switch into the asynchronous mode. Instead, the neurons may split into several subgroups (``cluster'') that fire alternatingly. Neurons within each group stay synchronized. An example of such a cluster state with two subgroups is illustrated in Fig. 8.9B. Action potentials produced by neurons from group 1 trigger group 2 neurons and vice versa. The population activity thus oscillates with twice the frequency of an individual neuron.

In general, there is an infinite number of different cluster states that can
be indexed by the number of subgroups. The length *T* of the
inter-spike interval for a single neuron and the number of subgroups
*n* in a cluster state are related by the threshold condition for spike
triggering (Kistler and van Hemmen, 1999; Chow, 1998),

- (T) = (k T/n) . |
(8.29) |

Stability is clarified by the Locking Theorem: A cluster state with

(t + k T/n) > 0 . |
(8.30) |

In Section 8.1 we have seen that the state of asynchronous firing in a SRM network is always unstable in the absence of noise. We now see that even if the fully locked state is unstable the network is not firing asynchronously but usually gets stuck in one of many possible cluster states. Asynchronous firing can only be reached asymptotically by increasing the number of subgroups so as to ``distribute'' the spike activity more evenly in time. Individual neurons, however, will always fire in a periodical manner. Nevertheless, increasing the number of subgroups will also reduce the amplitude of the oscillations in the input potential and the firing time of the neurons becomes more and more sensitive to noise. The above statement that asynchrony can only be reached asymptotically is therefore only valid in strictly noiseless networks.

A final remark on the stability of the clusters is in order. Depending on the form of the postsynaptic potential, the stability of the locked state may be asymmetric in the sense that neurons that fire too late are pulled back into their cluster, neurons that have fired to early, however, are attracted by the cluster that has just fired before. If the noise level is not too low, there are always some neurons that drop out of their cluster and drift slowly towards an adjacent cluster (Ernst et al., 1995; van Vreeswijk, 1996).

To illustrate the relation between the instability of the state of
asynchronous firing and cluster states, we return to the network of SRM_{0}
neurons with noisy reset that we have studied in Section 8.1. For low noise (
= 0.04), the asynchronous firing
state is unstable whatever the axonal transmission delay;
cf. Fig. 8.2. With an axonal delay of 2ms, asynchronous firing
is unstable with respect to an oscillation with . The population
splits into 3 different groups of neurons that fire with a period of about
8ms. The population activity, however, oscillates with a period of 2.7ms;
cf. Fig. 8.10A. With a delay of 1.2ms, the asynchronous
firing state has an instability with respect to so that the
population activity oscillates with a period of about 1.6ms. The population
splits into 5 diferent groups of neurons that fire with a period of about
8ms; cf. Fig. 8.10B.

Cambridge University Press, 2002

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