Subsections

• 1.6.1 Time-to-First-Spike
• 1.6.2 Phase
• 1.6.3 Correlations and Synchrony
• 1.6.4 Stimulus Reconstruction and Reverse Correlation

1.6 Spike Codes

In this section, we will briefly introduce some potential coding strategies based on spike timing.

1.6.1 Time-to-First-Spike

Let us study a neuron which abruptly receives a `new' input at time t₀. For example, a neuron might be driven by an external stimulus which is suddenly switched on at time t₀. This seems to be somewhat academic, but even in a realistic situation abrupt changes in the input are quite common. When we look at a picture, our gaze jumps from one point to the next. After each saccade, the photo receptors in the retina receive a new visual input. Information about the onset of a saccade would easily be available in the brain and could serve as an internal reference signal. We can then imagine a code where for each neuron the timing of the first spike after the reference signal contains all information about the new stimulus. A neuron which fires shortly after the reference signal could signal a strong stimulation, firing somewhat later would signal a weaker stimulation; see Fig. 1.12.

Figure 1.12: Time-to-first spike. The spike train of three neurons are shown. The third neuron from the top is the first one to fire a spike after the stimulus onset (arrow). The dashed line indicates the time course of the stimulus.

In a pure version of this coding scheme, each neuron only needs to fire a single spike to transmit information. (If it emits several spikes, only the first spike after the reference signal counts. All following spikes would be irrelevant.) To implement a clean version of such a coding scheme, we imagine that each neuron is shut off by inhibition as soon as it has fired a spike. Inhibition ends with the onset of the next stimulus (e.g., after the next saccade). After the release from inhibition the neuron is ready to emit its next spike that now transmits information about the new stimulus. Since each neuron in such a scenario transmits exactly one spike per stimulus, it is clear that only the timing conveys information and not the number of spikes.

A coding scheme based on the time-to-first-spike is certainly an idealization. In a slightly different context coding by first spikes has been discussed by S. Thorpe (Thorpe et al., 1996). Thorpe argues that the brain does not have time to evaluate more than one spike from each neuron per processing step. Therefore the first spike should contain most of the relevant information. Using information-theoretic measures on their experimental data, several groups have shown that most of the information about a new stimulus is indeed conveyed during the first 20 or 50 milliseconds after the onset of the neuronal response (Tovee and Rolls, 1995; Kjaer et al., 1994; Optican and Richmond, 1987; Tovee et al., 1993). Rapid computation during the transients after a new stimulus has also been discussed in model studies (Treves et al., 1997; Hopfield and Herz, 1995; van Vreeswijk and Sompolinsky, 1997; Tsodyks and Sejnowski, 1995). Since time-to-first spike is a highly simplified coding scheme, analytical studies are possible (Maass, 1998).

1.6.2 Phase

We can apply a code by 'time-to-first-spike' also in the situation where the reference signal is not a single event, but a periodic signal. In the hippocampus, in the olfactory system, and also in other areas of the brain, oscillations of some global variable (for example the population activity) are quite common. These oscillations could serve as an internal reference signal. Neuronal spike trains could then encode information in the phase of a pulse with respect to the background oscillation. If the input does not change between one cycle and the next, then the same pattern of phases repeats periodically; see Fig. 1.13.

Figure 1.13: Phase. The neurons fire at different phases with respect to the background oscillation (dashed). The phase could code relevant information.

The concept of coding by phases has been studied by several different groups, not only in model studies (Hopfield, 1995; Maass, 1996; Jensen and Lisman, 1996), but also experimentally (O'Keefe, 1993). There is, for example, evidence that the phase of a spike during an oscillation in the hippocampus of the rat conveys information on the spatial location of the animal which is not fully accounted for by the firing rate of the neuron (O'Keefe, 1993).

1.6.3 Correlations and Synchrony

We can also use spikes from other neurons as the reference signal for a spike code. For example, synchrony between a pair or many neurons could signify special events and convey information which is not contained in the firing rate of the neurons; see Fig. 1.14. One famous idea is that synchrony could mean `belonging together' (Milner, 1974; von der Malsburg, 1981). Consider for example a complex scene consisting of several objects. It is represented in the brain by the activity of a large number of neurons. Neurons which represent the same object could be `labeled' by the fact that they fire synchronously (von der Malsburg and Buhmann, 1992; Eckhorn et al., 1988; Gray and Singer, 1989; von der Malsburg, 1981). Coding by synchrony has been studied extensively both experimentally (Kreiter and Singer, 1992; Eckhorn et al., 1988; Gray and Singer, 1989; Singer, 1994; Engel et al., 1991b; Gray et al., 1989; Engel et al., 1991a) and in models (Eckhorn et al., 1990; Terman and Wang, 1995; Wang et al., 1990; König and Schillen, 1991; Gerstner et al., 1993a; von der Malsburg and Buhmann, 1992; Ritz et al., 1994; Wang, 1995; Schillen and König, 1991; Aertsen and Arndt, 1993). For a review of potential mechanism, see (Ritz and Sejnowski, 1997).

Figure 1.14: Synchrony. The upper four neurons are nearly synchronous, two other neurons at the bottom are not synchronized with the others.

More generally, not only synchrony but any precise spatio-temporal pulse pattern could be a meaningful event. For example, a spike pattern of three neurons, where neuron 1 fires at some arbitrary time t₁ followed by neuron 2 at time t₁ + $\delta_{{12}}^{}$ and by neuron 3 at t₁ + $\delta_{{13}}^{}$, might represent a certain stimulus condition. The same three neurons firing with different relative delays might signify a different stimulus. The relevance of precise spatio-temporal spike patterns has been studied intensively by Abeles (Abeles, 1994; Abeles et al., 1993; Abeles, 1991). Similarly, but on a somewhat coarse time scale, correlations of auditory and visual neurons are found to be stimulus dependent and might convey information beyond that contained in the firing rate alone (Steinmetz et al., 2000; deCharms and Merzenich, 1996).

1.6.4 Stimulus Reconstruction and Reverse Correlation

Let us consider a neuron which is driven by a time dependent stimulus s(t). Every time a spike occurs, we note the time course of the stimulus in a time window of about 100 milliseconds immediately before the spike. Averaging the results over several spikes yields the typical time course of the stimulus just before a spike (de Boer and Kuyper, 1968). Such a procedure is called a `reverse correlation' approach; see Fig. 1.15. In contrast to the PSTH experiment sketched in Section 1.5.2 where the experimenter averages the neuron's response over several trials with the same stimulus, reverse correlation means that the experimenter averages the input under the condition of an identical response, viz., a spike. In other words, it is a spike-triggered average; see, e.g., (de Ruyter van Stevenick and Bialek, 1988; Rieke et al., 1996). The results of the reverse correlation, i.e., the typical time course of the stimulus which has triggered the spike, can be interpreted as the `meaning' of a single spike. Reverse correlation techniques have made it possible to measure, for example, the spatio-temporal characteristics of neurons in the visual cortex (Eckhorn et al., 1993; DeAngelis et al., 1995).

Figure 1.15: Reverse correlation technique (schematic). The stimulus in the top trace has caused the spike train shown immediately below. The time course of the stimulus just before the spikes (dashed boxes) has been averaged to yield the typical time course (bottom).

With a somewhat more elaborate version of this approach, W. Bialek and his co-workers have been able to `read' the neural code of the H1 neuron in the fly and to reconstruct a time-dependent stimulus (Bialek et al., 1991; Rieke et al., 1996). Here we give a simplified version of their argument.

Results from reverse correlation analysis suggest, that each spike signifies the time course of the stimulus preceding the spike. If this is correct, a reconstruction of the complete time course of the stimulus s(t) from the set of firing times $\mathcal {F}$ = {t⁽¹⁾,...t⁽ⁿ⁾} should be possible; see Fig. 1.16.

As a simple test of this hypothesis, Bialek and coworkers have studied a linear reconstruction. A spike at time t^(f) gives a contribution $\kappa$(t - t^(f)) to the estimation s^est(t) of the time course of the stimulus. Here, t^(f) $\in$ $\mathcal {F}$ is one of the firing times and $\kappa$(t - t^(f)) is a kernel which is nonzero during some time before and around t^(f); cf. Fig. 1.16B. A linear estimate of the stimulus is

$$\sum_{{f=1}}^{n} \kappa$$ (1.12)

The form of the kernel $\kappa$ was determined through optimization so that the average reconstruction error $\int$dt[s(t) - s^est(t)]² was minimal. The quality of the reconstruction was then tested on additional data which was not used for the optimization. Surprisingly enough, the simple linear reconstruction (1.12) gave a fair estimate of the time course of the stimulus even though the stimulus varied on a time scale comparable to the typical interspike interval (Bialek et al., 1991; Bialek and Rieke, 1992; Rieke et al., 1996). This reconstruction method shows nicely that information about a time dependent input can indeed be conveyed by spike timing.

Figure 1.16: Reconstruction of a stimulus (schematic). A. A stimulus evokes a spike train of a neuron. The time course of the stimulus may be estimated from the spike train; redrawn after [Rieke et al., 1996]. B. In the framework of linear stimulus reconstruction, the estimation s^est(t) (dashed) is the sum of the contributions $\kappa$ (solid lines) of all spikes.