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Coherence-Detection by Spike-Synchronization

In operational terms, the process of coherence detection defined above is complicated, and the question arises how this process might be realized with available neural hardware. One answer to this question is simple, and closely connected to the behaviour of weakly coupled neural oscillators.

It is well known that a network of identical oscillators, coupled together by appropriate links, will synchronize under a wide variety of conditions [5,6,7,8,9,10]. The argument rests: 1) on the possibility of reducing most oscillator models under appropriate conditions to an equivalent model of phase oscillators; and 2) a generic condensation process leading towards synchronization for this type of oscillator.

Irrespective of the internal complexity of an oscillator, the temporal development of any isolated oscillatory unit can be described by a phasevariable, increasing from \( 0 \) to \( 2\pi \) in one oscillation of the unit. If several oscillators are coupled together, the description of the dynamics by a set of phasevariables might not be appropriate. However, if the coupling between the oscillatory units is weak enough, the dynamic of any single unit will only be slightly disturbed, compared to the uncoupled case. In this case a description of the whole system by a set of phasevariables is still acceptable.

If coupled phaseoscillators have the same oscillation frequency, they easily synchronize. However, the situation becomes more complicated if the oscillatory units have slightly different frequencies. The tendency for synchronization between any two oscillators is then counteracted by the difference between them. For a given weak coupling, there exist a maximum possible deviation between two oscillators to synchronize [5,6,7,11,12,13,14,15,16,17].

In summary, neural oscillators connected by weak coupling will display selective locking behaviour, depending on the magnitude of differences between them. Oscillators with minor differences in frequency will lock their signals, while oscillators with larger differences will not. Of course, such a selective locking behaviour is an exact replica of the search for the coherence cluster in the coherence-detection process defined above in equation (1).

While many neural circuits display oscillatory behaviour and might be used as basic computing elements for coherence detection, single neurons are used in this paper. The neurons are modeled as standard leaky integrate-and-fire neurons [see Appendix A.2.1 for details]. If supplied with an external current, these neurons spike regularly, with a rate depending on the strength of the input current.

Figure 2: Spike events in a pool of a 62 integrate-and-fire neurons coding disparity estimates along a common view direction. Spike trains are sorted according to their firing rate. If left uncoupled (A), no clear structure is visible in the data. With weak coupling (B, C), the neurons belonging to the coherence cluster quickly synchronize and appear as vertical blocks in the spike train display. For different stimuli, different groups of neurons are forming.
\resizebox* {0.9\columnwidth}{!}{\includegraphics{figs/sync_or_nosync/sync_or_nosync.eps}}

The neurons are assumed to drive other neurons with an exponentially decaying current, i.e., each spike supplies an input current \( \sim \, \exp (-t/\tau _{s}) \) to all postsynaptic neurons. Fig. 2 shows the temporal dynamics of a network of such integrate-and-fire neurons, with input currents reflecting the actual estimates of a group of disparity estimators (details in the Appendix). When left uncoupled (Fig. 2A), no obvious structure is visible in the spike trains of the neurons. By introducing a global, but weak synaptic coupling between the neurons, the network switches to a distinctively different behaviour (Fig. 2B). The pool of neurons quickly split into two separate sets, one a synchronous spiking cluster, which clearly shows up as sequence of vertical traces in the spike time diagram, and the rest of the pool, which stays asynchronous with the coherent cluster.

The attainment of synchronization under the weak-coupling paradigm is rapid, in this simulation within about 10 spike cycles of the coherence pool. Most important is that the decision of which neuron participates in the coherence cluster and which does not is stimulus-dependent: another stimulus situation changes the spike rates of the neurons, and in turn redistributes the synchronous cluster to other neurons within the pool (Fig. 2B, C).

The instantaneous stimulus value detected by the network is assumed to be coded by the spike frequency of the coherence cluster. To a first approximation, this frequency equals the average of the original, uncoupled spike frequencies of the units participating in the cluster, i.e., the frequencies before synchronization [6]. Thus, formula (2) of the coherence detection scheme is realized dynamically through the process of synchronization within the coherence cluster.

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Next: Marking by a synchronous Up: Construction by Coherence-Detection Previous: Grouping by Coherence-Detection