Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved

Abstract

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.

Functional Phase Response Curves: A Method for Understanding Synchronization of Adapting Neurons

“…Therefore synchrony was predicted to be unstable because there was no way to calculate the required slope for a perturbation from synchrony. At conductance values from 0.07 to 0.18 mS/cm 2 , near-leapfrog modes in which the order of firing changes every cycle (Maran and Canavier 2008;Oh and Matveev 2008) were observed to be bistable (Skinner et al 2005) with the antiphase mode. These leapfrog modes are not shown because they could not be predicted as the fixed points of the map described in the preceding session.…”

“…The protocols of obtaining PRCs for the adapting model neuron (Maran and Canavier 2008) were similar to the experimental method. The fPRCs were calculated at delays in increments of one hundredth of the period.…”

“…Previous work on neural oscillators has focused on two assumptions that allow PRCs to be used to predict the synchronization of mutually coupled oscillatory neurons. In the first case, the coupling is assumed to be weak (Ermentrout 2002;Jeong and Gutkin 2007;Mancilla et al 2007;Netoff et al 2005a;Preyer and Butera 2005;Rinzel and Ermentrout 1998), whereas in the second the coupling is assumed to be pulsatile (Canavier 2005;Maran and Canavier 2008;Netoff et al 2005b;Oprisan et al 2004;Pervouchine et al 2006). The approaches differ in the kind of PRC that is used and in their limitations.…”

Negative phase and extended spike-time response curve

“…The PRC-based analysis of the cell response involves describing the effect of perturbation as a change of the phase variable characterizing the state of the spiking cell, whereby the phase is always bounded on the interval [0, 1] or [0, 2π]. However, the extension of phase domain to negative values naturally arises when deriving phase return maps in the case of non-weak inhibition or larger networks coupled by three or more cells, as previously shown by Canavier and coworkers [1,2].In our previous work [3], we have shown that for some biophysical models of spiking cells, a one-dimensional phase reduction of a non-weakly perturbed limit cycle oscillator may require the extension of the phase variable defining the state of the oscillator to a multi-branched phase domain. Such multi-branched domain is most easily implemented by extending the [0, 1] phase interval to negative values.…”

“…The PRC-based analysis of the cell response involves describing the effect of perturbation as a change of the phase variable characterizing the state of the spiking cell, whereby the phase is always bounded on the interval [0, 1] or [0, 2π]. However, the extension of phase domain to negative values naturally arises when deriving phase return maps in the case of non-weak inhibition or larger networks coupled by three or more cells, as previously shown by Canavier and coworkers [1,2].…”