Functional phase response curves: a method for understanding synchronization of adapting neurons

Abstract

Phase response curves (PRCs) for a single neuron are often used to predict the synchrony of mutually coupled neurons. Previous theoretical work on pulse-coupled oscillators used single-pulse perturbations. We propose an alternate method in which functional PRCs (fPRCs) are generated using a train of pulses applied at a fixed delay after each spike, with the PRC measured when the phasic relationship between the stimulus and the subsequent spike in the neuron has converged. The essential information is the dependence of the recovery time from pulse onset until the next spike as a function of the delay between the previous spike and the onset of the applied pulse. Experimental fPRCs in Aplysia pacemaker neurons were different from single-pulse PRCs, principally due to adaptation. In the biological neuron, convergence to the fully adapted recovery interval was slower at some phases than that at others because the change in the effective intrinsic period due to adaptation changes the effective phase resetting in a way that opposes and slows the effects of adaptation. The fPRCs for two isolated adapting model neurons were used to predict the existence and stability of 1:1 phase-locked network activity when the two neurons were coupled. A stability criterion was derived by linearizing a coupled map based on the fPRC and the existence and stability criteria were successfully tested in two-simulated-neuron networks with reciprocal inhibition or excitation. The fPRC is the first PRC-based tool that can account for adaptation in analyzing networks of neural oscillators.

FIG. 1.

A: measurement of transient phase response curves (PRCs) with a spiking neuron of Aplysia californica. The period of the membrane potential (top trace) of the spiking neuron changed due to the injected current (bottom trace) changing at time ts after the onset of the spike. P₁ and P₂ illustrate the period of the perturbed cycle and of the following cycle, respectively. B: measuring functional PRCs (fPRCs) in an experiment using spiking neurons of Aplysia californica with excitatory input with parameters of maximum conductance g_syn = 0.04 μS and time constant τ = 10 ms. The distance between the dots above the voltage trace indicates the interspike intervals (ISIs) along the process of the experiment. The bottom trace indicates the time course of the current injected into the neuron; the blue trace illustrates the unperturbed neuron intrinsically firing action potentials; the red trace illustrates the periodic perturbations to make an fPRC measurement; and the green traces show the intrinsic firing of the unperturbed neuron after the stimulation is turned off. C: time sequence of ISIs for the specific fixed-delay time in B. The blue and green dots indicate the ISIs before and after repetitive stimuli; the red dots indicate the ISI being perturbed; the cyan star indicates the value of the fixed-delay time and the time when the repetitive stimulus is turned on.

FIG. 2.

Time series of ISI due to repetitive inhibitory (A) and excitatory (B) stimulation for a spiking neuron in Aplysia. Traces were reordered by the fixed-delay time, which occurred in random order. Legend is similar to that of Fig. 1C.

FIG. 3.

Illustration of comparing fPRCs (stars) with single-pulse first-order PRCs (f₁) (dots), second-order PRCs (f₂) (open circles), and f₁ + f₂ (squares) under the same experimental conditions for larger magnitude of inhibitory (A1) and excitarory (B1) input and smaller magnitude of inhibitory (A2) and excitarory (B2) input on 4 distinct neurons.

FIG. 4.

A and B: fixed-delay protocol for inhibition (A) and excitation (B). The light blue stars indicate the fixed delay. The intrinsic period in both cases is 60 ms. The synaptic conductance was 0.16 mS/cm². The stabilized network period while the delay is on is indicated by P_n. The estimated modified intrinsic period immediately after the fixed-delay protocol is turned off is indicated by P_m. To illustrate the method only 10 different delays are shown. C and D: phase resetting curves. The fPRC for inhibition (C) and excitation (D) is compared with first-, second-, and third-order PRCs as well as to the sum of these components.

FIG. 5.

Using the fPRC to predict phase locking in a 2-neuron network. A1: presumed firing patterns. A2: for the functional PRC, an input perturbation is applied with a fixed delay δ after the onset of spike. A3: in the closed-loop 2-neuron network, it is assumed that in a phase-locked mode, the recovery time for each neuron can be calculated as a function of δ from A2, which is equated with the stimulus time. B: the fPRC is used by setting the fixed delay between each spike and the perturbation equal to the stimulus interval ts for each neuron, then tabulating the recovery interval tr to the next spike in a 1:1 locked mode as a function of the delay. There are gaps in the function because a stable 1:1 locking was not established after 50 fixed-delay pulses at the missing points. The intersection marked by a circle predicts an antiphase mode with equal intervals of 10.875 ms. The parameters used are g_syn = 0.16 mS/cm², E_syn = 0 mV, I_app = 0.5 μA/cm², and τ_syn = 1 ms (star means when n goes to ∞). C and D: comparing predictions using PRCs and fPRCs. Red crosses indicate the observation; green circles indicate fPRC prediction, whereas blue circles indicate single-pulse PRC prediction. The Wang–Buzsáki (1996) model was used with E_syn = −75 for inhibition (C) and 0 mV for excitation (D).