4.5 Application: Coding by Spikes

Formal spiking neuron models allow a transparent graphical discussion of various coding principles. In this section we illustrate some elemantary examples.

We have seen in Chapter 1.4 that the time of the first spike can
convey information about the stimulus. In order to construct a simple
example, we consider a single neuron *i* described by the spike response
model SRM_{0}. The neuron receives spikes from *N* presynaptic neurons *j*
via synaptic connections that have all the same weight *w*_{ij} = *w*. There
is no external input. We assume that the last spike of neuron *i* occurred
long ago so that the spike after-potential in (4.42) can be
neglected.

At
*t* = *t*^{pre}, *n* < *N* presynaptic spikes are
simultaneously generated and produce a postsynaptic potential,

A postsynaptic spike occurs whenever

which is a function of

4.5.0.2 Phase Coding

Phase coding is possible if there is a periodic background signal that can
serve as a reference. We want to show that the phase of a spike contains
information about a static stimulus *h*_{0}. As before we take the model
SRM_{0} as a simple description of neuronal dynamics. The periodic background
signal is included into the external input. Thus we use an input potential

where

Let us consider a single neuron driven by (4.106). The membrane
potential of a SRM_{0} neuron is, according to (4.42) and (4.46)

As usual denotes the time of the most recent spike. To find the next firing time, Eq. (4.107) has to be combined with the threshold condition

For a given period

4.5.0.3 Correlation coding

Let us consider two uncoupled neurons. Both receive the same constant
external stimulus
*h*(*t*) = *h*_{0}. As a result, they fire regularly with
period *T* given by
(*T*) = *h*_{0} as can be seen directly from
Eq. (4.108) with *h*_{1} = 0. Since the neurons are not coupled, they need not
fire simultaneously. Let us assume that the spikes of neuron 2 are shifted by
an amount with respect to neuron 1.

Suppose that, at a given moment
*t*^{pre}, both neurons receive input from
a common presynaptic neuron *j*. This causes an additional contribution
(*t* - *t*^{pre}) to the membrane potential. If the synapse is
excitatory, the two neurons will fire slightly sooner. More importantly, the
spikes will also be closer together. In the situation sketched in Fig.
4.27 the new firing time difference
is reduced,
< . In later chapters, we will analyze this phenomenon
in more detail. Here we just note that this effect would allow us to encode
information using the time interval between the firings of two or more
neurons.

In the previous paragraphs we have studied how a neuron can encode information
in spike timing, phase, or correlations. We now ask the inverse question,
viz., how can a neuron read out temporal information? We consider the
simplest example and study whether a neuron can distinguish synchronous from
asynchronous input. As above we make use of the simplified neuron model
SRM_{0} defined by (4.42) and (4.43). We will show that
synchronous input is more efficient than asynchronous input in driving a
postsynaptic neuron.

To illustrate this point, let us consider an kernel of the form

We set

Let us consider a neuron *i* which receives input from 100 presynaptic neurons
*j*. Each presynaptic neuron fires at a rate of 10 Hz. All synapses have the
same efficacy *w* = 1. Let us first study the case of asynchronous input.
Different neurons fire at different times so that, on average, spikes arrive
at intervals of
*t* = 1 ms. Each spike evokes a postsynaptic potential
defined by (4.109). The total membrane potential of neuron *i* is

If neuron

u_{i}(t) (s) ds = = 10 mV . |
(4.111) |

If the firing threshold of the neuron is at = 20 mV the neuron stays quiescent.

Now let us consider the same amount of input, but fired synchronously at
*t*_{j}^{(f)} = 0, 100, 200,...ms. Thus each presynaptic neuron fires as before at 10
Hz but all presynaptic neurons emit their spikes synchronously. Let us study
what happens after the first volley of spikes has arrived at *t* = 0. The
membrane potential of the postsynaptic neuron is

where

We will return to the question of coincidence detection, i.e., the distinction between synchronous and asynchronous input, in the following chapter. For a classical experimental study exploring the relevance of temporal structure in the input, see Segundo et al. (1963).

In neurons with a spatially extended dendritic tree the form of the
postsynaptic potential depends not only on the type, but also on the location
of the synapse; cf. Chapter 2. To be specific, let us
consider a multi-compartment integrate-and-fire model. As we have seen above
in Section 4.4, the membrane potential *u*_{i}(*t*) can
be described by the formalism of the Spike Response Model. If the last output
spike is long ago, we can neglect the refractory kernel
and the membrane potential is given by

u_{i}(t) = w_{ij}(t - t_{j}^{(f)}). |
(4.113) |

cf. Eq. (4.90). The subscript

Cambridge University Press, 2002

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