3. Signal processing in mean-driven neurons 17
3.4. Phase dynamics
Figure 3.4.:Sketch of the relation between membrane voltage and input for different spike onset bifurcations. Fixed point voltage in violet, limit cycle (LC) maximal and minimal voltage in green. Straight lines denote linearly stable dynamics, dashed lines linear unstable dynamics. A:
Saddle-node on invariant cycle bifurcation as typical for “type-I” neurons. B: Saddle-homoclinic orbit bifurcation (for an example, see Fig. 3.2 and Fig. 3.3 at the highest capacitance value). C:
Bifurcation structure as observable below the Bogdanov-Takens point in the second publication, Figure 6. D: Subcritical Hopf bifurcation (“type-II”) as in the original Hodgkin-Huxley model [73].
signal transmission in the first publication.
3.4. Phase dynamics
While conductance-based neuron models allow for a detailed analysis of neuronal dynamics, depending on a variety of biologically inspired parameters, particular as-sumptions allow to capture the essential spike dynamics of the (potentially multi-dimensional) dynamical system already with a single dynamical variable, thephase ϕ. Thephase reductionis in general applicable to (stable) limit cycle dynamics that are weakly perturbed [96]. How weak the perturbation has to be depends on the limit cycle itself, more precisely on its attractiveness. The more attractive the limit cycle is in its immediate environment, the stronger the perturbations that are still admissible.
As a result, the perturbation hardly influences the shape of the limit cycle, but only advances or delays the limit cycle dynamics in time.
The phase reduction can be applied to neurons that spike continuously in time.
Perturbations,i.e., input from synaptic coupling or stimulation electrode, are required to advance or delay the occurrence of the next spike (and only the next spike, to be strict), while leaving the spike shape mostly undisturbed.
These assumptions form the starting point for the analysis in the second project. The dynamics can then be captured with a considerably simpler equation compared to the original system in Eq. 3.1, which is given for the phaseϕas
ϕ˙ =1/T+Z(ϕ)s(t), (3.3) with limit cycle periodT, a time-dependent inputs(t)andphase-response curve(PRC) Z(ϕ), which is determined from the original system Eq. 3.1 [96]. The definition of the phase is translational invariant in time. As a convention, integer crossings of the phase are identified with the voltage maximum of subsequent spikes. In real time, spikes are correspondingly observed at timesTϕforϕ∈Z. For the unperturbed case, i.e., s(t) = 0, the phase is just a scaled version of the time ϕ = t/T. With nonzero stimulus, the multiplication with the phase-response curve in Eq. 3.3 translates the stimulus into an advance or delay in the occurrence time of the next spike (phase advance/delay). The phase-response curve gives the phase advance or delay for a delta perturbation as sketched in Fig. 3.5, and the so calledinfinitesimalphase-response curve used in the following has a unit that is in Hz/[stim], where [stim] denotes the unit of the stimuluss(t)[39]. The phase-response curve is also known by the names phase resetting curve, phase sensitivity or phase susceptibility. For mathematical details on the phase reduction and the phase-response curve please refer to the classical book by Kuramoto [96], or for an overview in the context of neuroscience to Ermentrout and Terman [38], as well as the references given in those.
Figure 3.5.:The phase-response curve measures the phase advance (or delay) of the next spike in response to a delta perturbation.
The phase-response curve is, via its mathematical properties as solution to the adjoint equation of the dynamics, tightly related to the limit cycle period and the flow field around the limit cycle [142]. In the second publication, the membrane capacitance is changed, which directly affects the relaxation time of the voltage dynamics (Eq. 3.1).
This changes the amplitude and the direction of the flow field in the phase space of voltage and each gating variable, and hence affects the phase-response curve in a direct way, both via the limit cycle period and the flow field. The phase-response curve changes particularly drastic around the loop bifurcation. The saddle-node-loop bifurcation not only deforms the flow field around the limit cycle, but the orbit flip lets the limit cycle (or at least half of it) jump to a new location, with a substantially different flow field than before. At the saddle-node-loop bifurcation, the orbit flips from the semistable manifold to the strongly stable manifold for the approach of the
saddle-3.4. Phase dynamics node fixed point. This speeds up half of the limit cycle such that it can be effectively neglected in the phase description. This is also reflected in the phase-response curve, which is practically halved. This has drastic consequences for synchronization and other coding properties. A more detailed presentation of these points can be found in the second publication and Sec. 8.3.
3.4.1. Synchronization inferred from individual cells
The analysis in the second publication uses the phase-response curves around the saddle-node-loop bifurcation to derive drastic changes in the ability of cells to syn-chronize. What follows is an intuitive presentation of how synchronization on the network level can be inferred, via the phase-response curve, from properties of a single cell. The full mathematical derivation of the relation between synchronization and phase-response curve can be found in Kuramoto [96].
For a repetitively firing neuron, the phase-response curve measures how much a small input influences the timing of the next spike. When two neurons are synaptically coupled, they mutually affect each other’s spiking. The interaction can synchronize the spiking,i.e., the spikes occur at a fixed phase relation (the spike of one neuron occurs always at the same phase of the other). Note that this definition of synchronization (so-calledphase synchronization) also includes what is sometimes calledanti-synchronization, for which spiking alternates between both neurons. While anti-synchronization may not be easily recognized when observing more than a few units, in-phase synchro-nization has been proposed as a mechanism to link, for example, spatially separated representation of the same object [56]. Synchronization may also underlie rhythm gen-eration in the hippocampus, important for memory [24], and has been related to several diseases such as schizophrenia, Parkinson’s or epilepsy [168]. Spike synchronization to a common input, which is also facilitated around the saddle-node-loop bifurcation (see Sec. 8.3), is for example used to encode odors in the olfactory system [13, 77].
Figure 3.6.:Top: Spike times of two neurons with different but similar inter-spike-intervals.
Bottom: Spike times of the same two neurons when coupled with delta-synapses such that the interaction is ruled by the asymmetric phase-response curve (black with red arrows). The symmetric phase-response curve would not lead to a constant phase differenceδ, as both spikes would be advanced similarly (gray arrow), preserving the difference in the inter-spike-intervals.
In the example of Fig. 3.6, the two neurons synchronize when their mutual influence is asymmetric. Synchronization is reached when the neuron with the higher baseline firing rate is able to advance the spiking of the neuron with the lower baseline firing rate
more thanvice versa. With excitatory coupling, the inter-spike-interval of each neuron will be shorter than the uncoupled inter-spike-interval due to the input it receives from the other neuron. A possible common inter-spike-interval duration is therefore shorter than both baseline inter-spike-interval. For synchronization, one neuron fires at a constant time delayδbefore the other neuron. That implies that the first neuron receives the perturbation from the second neuron at a timeδafterthe first spikes, while the second neuron receives the perturbation from the first neuron at a timeδ before the second spikes. A symmetric phase-response curve implies in such a symmetric coupling that the change in the inter-spike-intervals of both neurons is the same, and thus the difference between the baseline inter-spike-intervals cannot be reduced by the coupling. However, if the phase-response curve has an asymmetric component, this asymmetry can balance the asymmetry in the baseline inter-spike-intervals, and thereby allows for synchronization. Also in general, the asymmetry of the phase-response curve is related to the maximal frequency detuning (difference in firing rate) between two neurons that still allows them to synchronize. Assuming delta-synapses (coupling where a spike in one neurons induces an instantaneous increase in voltage in the other), the amplitude of the odd part of the phase-response curve is also related to their ability for synchronization in larger networks. In summary, (twice) the odd part of the phase-response curve corresponds to theentrainment rangeof two coupled oscillators,i.e., the maximal phase difference (resulting from different baseline firing rates) that still allows for phase synchronization between the two. For details see Kuramoto [96].
While it seems at first glance counterintuitive to infer synchronization from a single cell, analog reasoning is used in everyday life. For example, for the Olympic discipline
“synchronized swimming”, the performance of the swimmers is evaluated based on their synchrony. While normally the synchrony is assessed from the actual performance of the whole group, it could, in principle, be already predicted based on the precision of one of the performers: The more professional each individual, the better the individual can be expected to observe the co-swimmers during his or her own action, and the more synchronous the group may perform. In a comparable way, synchronization of interacting neurons can also be estimated from the individual characteristics of the neurons.