Subsections

• • 4.5.0.1 Time-to-First-Spike
  • 4.5.0.2 Phase Coding
  • 4.5.0.3 Correlation coding
  • 4.5.0.4 Decoding: Synchronous versus asynchronous input
  • 4.5.0.5 Example: Spatio-temporal summation

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.

4.5.0.1 Time-to-First-Spike

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 SRM₀. 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 $\eta$ in (4.42) can be neglected.

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

$$\epsilon$$ (4.104)

A postsynaptic spike occurs whenever u_i reaches the threshold $\vartheta$. We consider the firing time t_i^(f) of the first output spike,

$$\vartheta$$ (4.105)

which is a function of n. A larger numer of presynaptic spikes n results in a postsynaptic potential with a larger amplitude so that the firing threshold is reached earlier. The time difference t_i^(f) - t^pre is hence a measure of the number of presynaptic pulses. To put it differently, the timing of the first spike encodes the strength of the input; cf. Fig. 4.25.

Figure 4.25: Time-to-first-spike. The firing time t^f encodes the number n₁ or n₂ of presynpatic spikes which have been fired synchronously at t^pre. If there are less presynaptic spikes, the potential u rises more slowly (dashed) and the firing occurs later. For the sake of simplicity, the axonal delay has been set to zero; taken from Gerstner (1998).

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 h₀. As before we take the model SRM₀ as a simple description of neuronal dynamics. The periodic background signal is included into the external input. Thus we use an input potential

$$\left(\vphantom{2\pi\,{t\over T}}\right. \pi {t\over T} \left.\vphantom{2\pi\,{t\over T}}\right)$$ (4.106)

where h₀ is the constant stimulus and h₁ is the amplitude of the T-periodic background; cf. Eq. (4.46).

Let us consider a single neuron driven by (4.106). The membrane potential of a SRM₀ neuron is, according to (4.42) and (4.46)

$$\eta \hat{{t}}$$ (4.107)

As usual $\hat{{t}}$ 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 u(t) = $\vartheta$. We are interested in a solution where the neuron fires regularly and with the same period as the background signal. In this case the threshold condition reads

$$\vartheta \eta \left(\vphantom{2\pi {\hat{t}\over T}}\right. \pi {\hat{t}\over T} \left.\vphantom{2\pi {\hat{t}\over T}}\right)$$ (4.108)

For a given period T, the left-hand side has a fixed value and we can solve for $\varphi$ = 2$\pi$$\hat{{t}}$/T. There are two solutions but only one of them is stable. Thus the neuron has to fire at a certain phase $\varphi$ with respect to the external signal. The value of $\varphi$ depends on the level of the constant stimulation h₀. In other words, the strength h₀ of the stimulation is encoded in the phase of the spike. In Eq. (4.108) we have moved $\eta$ to the left-hand side in order to suggest a dynamic threshold interpretation. A graphical interpretation of Eq. (4.108) is given in Fig. 4.26.

Figure 4.26: Phase coding. Firing occurs whenever the total input potential h(t) = h₀ + h₁ cos(2$\pi$t/T) hits the dynamic threshold $\vartheta$ - $\eta$(t - $\hat{{t}}$) where $\hat{{t}}$ is the most recent firing time; cf. Fig. 1.11. In the presence of a periodic modulation h₁$\ne$ 0, a change $\Delta$h₀ in the level of (constant) stimulation results in a change $\Delta$$\varphi$ in the phase of firing; taken from Gerstner (1998).

4.5.0.3 Correlation coding

Let us consider two uncoupled neurons. Both receive the same constant external stimulus h(t) = h₀. As a result, they fire regularly with period T given by $\eta$(T) = h₀ as can be seen directly from Eq. (4.108) with h₁ = 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 $\delta$ 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 $\epsilon$(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 $\tilde{{\delta}}$ is reduced, $\tilde{{\delta}}$ < $\delta$. 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.

Figure 4.27: The firing time difference $\delta$ between two independent neurons is decreased to $\tilde{{\delta}}$ < $\delta$, after both neurons receive a common excitatory input at time t^pre; taken from Gerstner (1998).

4.5.0.4 Decoding: Synchronous versus asynchronous input

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 SRM₀ 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 $\epsilon$ kernel of the form

$$\epsilon_{0}^{} {s\over \tau} \left(\vphantom{-{s\over \tau}}\right. {s\over \tau} \left.\vphantom{-{s\over \tau}}\right) \Theta$$ (4.109)

We set J = 1 mV and $\tau$ =10 ms. The function (4.109) has a maximum value of J/e at s = $\tau$. The integral over s is normalized to J$\tau$.

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 $\Delta$t = 1 ms. Each spike evokes a postsynaptic potential defined by (4.109). The total membrane potential of neuron i is

$$\eta \hat{{t}}_{i}^{} \sum_{j}^{} \sum_{{t_j^{(f)}}}^{} \epsilon_{0}^{}$$ u_i(t) =

$$\approx \eta \hat{{t}}_{i}^{} \sum_{{n=0}}^{\infty} \epsilon_{0}^{} \Delta$$ (4.110)

If neuron i has been quiescent in the recent past ( t - $\hat{{t}}_{i}^{}$$\to$$\infty$), then the first term on the right-hand side of (4.110) can be neglected. The second term can be approximated by an integral over s, hence

$$\approx {w\over \Delta t} \int_{0}^{\infty} \epsilon_{0}^{} {w\,J\,\tau\over \Delta t}$$ (4.111)

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

Figure 4.28: Potential u of a postsynaptic neuron which receives input from two groups of presynaptic neurons. A. Spike trains of the two groups are phase shifted with respect to each other. The total potential u does not reach the threshold. There are no output spikes. B. Spikes from two presynaptic groups arrive synchronously. The summed EPSPs reach the threshold $\vartheta$ and cause the generation of an output spike.

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

$$\eta \hat{{t}}_{i}^{} \epsilon_{0}^{}$$ (4.112)

where N = 100 is the number of presynaptic neurons. If the postsynaptic neuron has not been active in the recent past, we can neglect the refractory term $\eta$ on the right-hand side of Eq. (4.112). The maximum of (4.112) occurs at t = $\tau$ = 10 ms and has a value of wNJ/e $\approx$ 37 mV which is above threshold. Thus the postsynaptic neuron fires before t = 10 ms. We conclude that the same number of input spikes can have different effects depending on their level of synchrony; cf. Fig. 4.28.

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).

4.5.0.5 Example: Spatio-temporal summation

Figure 4.29: Sensitivity to temporal order of synaptic inputs on a dendrite. A. A neuron is stimulated by three synaptic inputs in a sequence that starts at the distal part of the dendrite and ends with an input close to the soma. Since the EPSP caused by the distal input has a longer rise time than that generated by the proximal input, the EPSPs add up coherently and the membrane potential reaches the firing threshold $\vartheta$. B. If the temporal sequence of spike inputs is reversed, the same number of input spikes does not trigger an action potential (schematic figure).

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 $\hat{{t}}_{i}^{}$ is long ago, we can neglect the refractory kernel $\eta_{i}^{}$ and the membrane potential is given by

$$\sum_{j}^{} \sum_{f}^{} \epsilon_{{ij}}^{}$$ (4.113)

cf. Eq. (4.90). The subscript ij at the $\epsilon$ kernel takes care of the fact that the postsynaptic potential depends on the location of the synapse on the dendrite. Due to the low-pass characteristics of the dendrite, synaptic input at the tip of the dendrite causes postsynaptic potentials with a longer rise time and lower amplitude than input directly into the soma. The total potential u_i(t) depends therefore on the temporal order of the stimulation of the synapses. An input sequence starting at the far end of the dendrite and approaching the soma is more effective in triggering an output spike than the same number of input spikes in reverse order; cf. Fig. 4.29.