Phase Response Properties of Half-Center. Oscillators

Transcription

1 Phase Response Properties of Half-Center Oscillators Jiawei Calvin Zhang Timothy J. Lewis Department of Mathematics, University of California, Davis Davis, CA 95616, USA June 17, 212 Abstract We examine the phase response properties of half-center oscillators (HCOs) that are modeled by a pair of Morris-Lecar neurons strongly and reciprocally connected by fast inhibitory synapses. We find that the two basic mechanisms for half-center oscillations, release and escape, give rise to strikingly different phase response curves (PRCs). Release-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the active to the suppressed (inhibited) state. This is manifested in the PRC as a large dominant negative peak (i.e., phase delays) at corresponding phases. On the other hand, escapetype HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the suppressed to the active state, and the PRC is dominated by a large positive peak (i.e., phase advances) at corresponding phases. We identify the dynamical mechanisms underlying the shapes of the PRCs by analyzing the phase-space structure of the fast-slow HCO system. Finally, we show that the different phase response 1

2 properties of the release-type and escape-type HCOs can lead to fundamentally different phase-locking dynamics and frequency modulation. Keywords: central pattern generators (CPGs), half-center oscillators (HCOs), phase response curves (PRCs), phase plane analysis, release and escape mechanisms 1 Introduction Central pattern generators (CPGs) are specialized neuronal circuits in the central nervous system that produce rhythmic activity in the absence of afferent feedback or rhythmic input [11, 24]. Almost all rhythmic movements of animals (walking, swimming, breathing, chewing, etc.) are programmed at least in part by CPGs. Because of the changing demands of an organism, CPGs must be able to adjust their activity to produce the appropriate motor behavior. CPGs must respond appropriately (e.g., shift their phase or regulate their frequency) to input from higher centers and sensory feedback. Furthermore, behaviors such as locomotion are controlled by a network of interconnected CPG modules that interact to produce the limb or segmented-body movements necessary for effective locomotion, and therefore CPG modules must adjust their phases in response to input from other modules in order to produce the properly coordinated activity [28, 3, 42]. This implies that a comprehensive understanding of CPG activity must include an understanding of how they respond to input [38, 6]. A useful way to characterize how oscillators respond to input is their phase response curves (PRCs) [33]. PRCs measure the phase shift of an oscillator in response to a brief perturbation as a function of the timing of the input. PRCs in conjunction with coupled oscillators theory have been widely used in neuroscience to understand how changes in intrinsic properties affect a neuron s frequency [35, 21], how neuronal oscillators phase-lock to external input [13, 8], 2

3 and how networks of neurons synchronize (e.g., [7, 1, 23]). In almost all cases, the oscillatory units are taken to be the individual neurons. Despite the fact that oscillations in neural systems often arise from network interactions, there has been very few theoretical studies on the phase response properties of networkbased oscillators (see however [16, 32, 44, 14]). Furthermore, coupled oscillator theory has been used extensively to study phase-locking in networks of CPGs, but almost all of these studies have used a highly idealized model for the coupling functions (i.e., the Kuramoto model) [3, 17, 47, 38] rather than PRCs from biophysically-based models. It remains unclear how the intrinsic properties of cells and the synaptic coupling combine to produce the phase response properties of network-based oscillators and CPGs in particular. In this article, we begin to explore the response properties of CPGs by examining the phase response characteristics of an idealized half-center oscillator (HCO) model. HCOs consist of two neurons (or neuronal populations) connected by reciprocal inhibition [1, 27]. This half-center organization is integral to many CPG circuits (e.g., [2, 31, 26, 4]). The reciprocal inhibition and some form of slow adaptation leads to antiphase oscillations in which the units alternate between an active state and a suppressed (inhibited) state. Two basic mechanisms for generating half-center oscillations have been described [45, 38]: release in which the oscillations are primarily controlled by the active cell releasing the suppressed cell from inhibition, and escape in which the oscillations are primarily controlled by the suppressed cell escaping from inhibition. We find that the release and escape mechanisms for Morris-Lecar-type HCOs give rise to strikingly different PRCs. We use geometric analysis of the phase portrait structure to explain how release and escape dynamics shape the PRCs, and we then give examples of how the shapes of the PRCs can lead to fundamentally different responses to external input. 3

4 2 Methods 2.1 Description of Half-Center Oscillator Model The model half-center oscillator (HCO) used in this study consists of two neurons that are strongly coupled by fast reciprocal inhibitory synapses. Specifically, we use the HCO model described by Skinner et al. [38]. The dynamics of each neuron are described by standard current balance equations based on the Morris-Lecar model [25]. The ionic currents in this model include a fastactivating inward current (the calcium current), an outward current with relatively slow activation kinetics (the potassium current), and a voltage independent leakage current. The strong inhibitory synapses are modeled using fast threshold modulation synapses [41]. The model equations are C dv i dt =I g Cam (V i ) ( V i E Ca ) gk n i ( Vi E K ) ( ) g L Vi E L Isyn,j i + I ext,i (t), (1) dn i dt = ɛ n ( ) n (V i ) n i, τ n (V i ) (2) where V i is the membrane potentials for cell i, n i is the gating variable for the potassium current in cell i (i, j =1, 2,i j). C is the membrane capacitance; g Ca,g K and g L are the maximal conductances and E Ca,E K and E L are the reversal potentials for the calcium, potassium and leakage currents, respectively; I is a bias current (which could be absorbed into E L ); I ext,i is an external current injected into cell i. The functions m (V )andn (V ) describe the voltage-dependent steady state activation curves for the calcium and potassium 4

5 conductances m (V )= 1 2 n (V )= 1 2 [ ( V Va 1+tanh [ 1+tanh )], (3) V b ( V Vc )], (4) V d and τn(v ) ɛ n is the time constant for potassium conductance activation, in which ( τ n (V i )=cosh 1 V V ) c. (5) 2V d The activation dynamics for the calcium conductance (g Ca m (V )) is taken to be an instantaneous function of the membrane potential. The level of activation of the potassium conductance (n) is modeled as a dynamic variable to capture its relatively slow response to changes in the membrane potential, which can be controlled by scaling the factor ɛ n. The fast synaptic current imposed by neuron j onto neuron i is I syn,j i = g syn s (V j ) ( V i E syn ), (6) where g syn and E syn are the maximal conductance and the reversal potential for the synaptic currents, respectively. The synaptic conductances are assumed to have fast dynamics and are taken to be instantaneous functions of the membrane potential of the presynaptic neuron (V pre ). Specifically, the level of activation of the synapses is given by the monotonically increasing sigmoid function s (V pre )= 1 2 [ ( Vpre η )] syn 1+tanh, (7) k syn where η syn is the membrane potential at which the synaptic conductance is halfmaximal, and k syn determines the steepness of the synaptic activation curve s (V ). The synaptic activation curve is assumed to be steep, i.e., k syn is small. 5

6 Therefore, we interpret η syn as the synaptic threshold. All simulations of the Morris-Lecar-based HCO model were performed using the same parameter values as Skinner et al. [38] except for ɛ n and k syn. (We relax their assumptions that the parameters ɛ n and k syn are extremely small). Unless otherwise stated, C =1μ F/cm 2, g Ca =.15 ms/cm 2, E Ca=1 mv, g K =.2 ms/cm 2, E K = 8 mv, g L =.5 ms/cm 2, E L = 5 mv, g syn =.1 ms/cm 2, E syn = 8 mv, V a =mv,v b =15mV,V c =mv, V d =15mV,I =.8 μa/cm 2, ɛ n =.5, k syn =2mV,andη syn is defined in each example. I ext,i =(i =1, 2) except when studying the effects of small perturbations or periodic inputs to the HCO model. We stress that the results of this study do not depend on the specific combination of parameters used or the implementation of the Morris-Lecar-type model. 2.2 Phase Response Curves A phase response curve (PRC) quantifies the sensitivity of an oscillator at different phases in its cycle [35]. Specifically, the PRC of a neuronal oscillator measures its phase shift in response to a brief current pulse as a function of the phase at which the pulse is delivered. The convention adopted in this article is that a phase advance is indicated by a positive value on the PRC, whereas a phase delay is indicated by a negative value. If the current pulse amplitude is sufficiently small and its duration is sufficiently brief, then the phase shift will be proportional to the total charge injected during the pulse. In this case, when the phase shift is normalized by the total charge and membrane capacitance, one obtains the infinitesimal PRC, which has units of time per unit membrane potential (msec/mv). In what follows, we use PRC to refer to the infinitesimal PRC, which we denote by Z(θ) whereθ is the phase of the oscillator (in units of time), and T is the period of the oscillation. For ease of comparison, 6

7 the PRCs showed in this article are 1 T Z( θ T ) (in units of 1/mV), in which the phase θ and the phase response Z(θ) are both normalized by the period T. In neuroscience, PRCs typically are determined for single oscillatory neurons, but they can also be used to characterize the response properties of neural systems in which oscillations arise from network properties. Note that despite a network-based oscillator contains more than one cell, the oscillator has a single phase, θ. However, perturbations can be delivered to each cell in the network, and hence there is a different PRC for each neuron. For our HCO model, which consists of two identical neurons oscillating in anti-phase, the PRCs for each neuron are identical except for a phase shift of half a period (T/2). We define the PRC for cell 1 to be Z(θ), referred by Z 1 (θ), and then the PRC for cell 2 is Z(θ + T/2), referred by Z 2 (θ). PRCs for individual neurons and neuronal networks can be obtained directly by applying current pulses at many different times in the oscillator s cycle (e.g., [23, 39]). However, PRCs for mathematical models can be computed more efficiently by solving the adjoint equations for the full model equations linearized around the system s T-periodic limit cycle. The PRC is the appropriately normalized T-periodic solution of the adjoint equations. The PRCs for the HCO model (Eqn. 1 2) presented in this article are computed using this so-called adjoint method as described by Williams and Bowtell [46]. However, example PRCs were checked against PRCs that were obtained by using the direct method. 2.3 Phase Models Besides characterizing the phase-dependent sensitivity of the HCO, the PRC can be used to examine the oscillator s phase-locking dynamics in response to periodic external input. Using the Theory of Weakly Coupled Oscillators 7

8 [19, 22, 7], the dynamics of the HCO can be reduced to the consideration of a phase model, in which the state of the oscillator is described entirely by its phase θ(t). The phase can be decomposed into two parts: θ(t) = ( t + φ(t) ) mod T,wheret describes the change in the phase of the oscillator in the absence of the external input, and φ(t), which is called the relative phase, quantifies the phase shift that results from the external input. If the effect of the T -periodic external input on the phase of the oscillator is relatively small compared to that of the intrinsic dynamics of the oscillator, then the change in φ(t) isvery small (negligible) over a single period, i.e., φ(t) is approximately constant over a single period. However, these small changes in φ(t) can accumulate over many periods to produce significant phase shifts. This slow-time evolution of φ(t) for the HCO is captured by the phase equation dφ dt = 1 T T Z 1 (s + φ) I ext,1(s) ds + 1 T C T Z 2 (s + φ) I ext,2(s) ds C =H 1 (φ)+h 2 (φ) H(φ), (8) where H i (φ) = 1 T T Z i(s + φ) Iext,i(s) C ds. The so-called interaction function H i (φ) quantifies the average rate of change of the relative phase of the HCO over a period T that results from the external input into cell i at any particular relative phase (φ constant within a single period). The phase equation (Eqn. 8) is a first-order autonomous differential equation that can be easily analyzed. Specifically, the zeros of H(φ) correspond to the steady states of the phase model and therefore the phase-locked states of the oscillator to the external T -periodic input I ext,i (t). These phase-locked states are stable if dh(φ) dφ <. Note that the above mathematical framework can be extended to accommodate the case in which the external forcing is not strictly T -periodic and to study the phase-locking properties of coupled oscillator networks. Details of the Theory 8

9 of Weakly Coupled Oscillators can be found in the recent review by Schwemmer and Lewis [35]. 3 Results In this section, we first review the basic dynamical mechanisms of half-center oscillations and show that the phase response curves (PRCs) for the releasetype and the escape-type half-center oscillators (HCOs) have strikingly different shapes. We then describe how the dynamics of the release and escape mechanisms shape the PRCs of HCOs. Finally, we use some examples to show that the shapes of the HCO s PRC can be used to understand how external perturbations affect a HCO s frequency of oscillation and its phase-locking properties. 3.1 Mechanisms for Oscillations in HCOs: Release and Escape A HCO consists of two non-oscillatory cells that are coupled by mutual inhibition. Fast synaptic inhibition and slow activiation and deactivation kinetics of the potassium currents leads to anti-phase oscillations in which the cells transition between an active state and a suppressed (inhibited) state. If the cells transition between the active state and the suppressed state is triggered by changes in the active cell, the underlying mechanism is called release ; if the transition is triggered by changes in the suppressed cell, it is called escape. In this subsection, we describe the dynamics for these two types of HCOs. For simplicity, the examples are idealized cases in which the cells in the HCO are assumed to undergo relaxation-oscillator-like dynamics (i.e., <ɛ n 1) and the synapses connecting the cells are assumed have a sharp threshold for the activation (i.e., k syn small) [38]. However, we will show that the basic behavior 9

10 and main conclusions in the article carry over to less restrictive cases. For more general and detailed descriptions of HCO dynamics and the release and escape mechanisms, see Wang and Rinzel [45], Skinner et al. [38] and Daun et al. [6] Release Mechanism We first describe the dynamics of the HCO under the release mechanism. Fig. 1a plots the membrane potentials V i and the gating variables for the outward K + current n i as functions of phase for cell 1 and cell 2 during steady state oscillations. At phase θ =, cell 1 has just made a transition from the suppressed state (the inhibited state) to the active state (the free state). Its membrane potential V 1 is at a maximal value, which is above the synaptic threshold η syn (indicated by the horizontal dotted line in the top figure in Fig. 1a), and therefore cell 1 is inhibiting cell 2. Cell 2 has just been driven from the active state to the suppressed state by cell 1, and V 2 is at a minimal membrane potential. As the cycle progresses, the outward K + current in cell 2 slowly deactivates (i.e., n 2 slowly decreases), leading to slow depolarization of cell 2. On the other hand, the outward K + current slowly activates in cell 1 (i.e., n 1 slowly increases), leading to a slow decrease in V 1. If it were isolated, cell 1 would asymptotically approach an active steady state around 13 mv. However, as V 1 decreases toward this steady state, it crosses η syn at around phase θ =.5T (normalized phase.5), which turns off the inhibitory synapse from cell 1 to cell 2. Upon its release from inhibition, cell 2 s membrane potential V 2 rapidly depolarizes and rises above η syn. Cell 1 is now inhibited, and V 1 rapidly hyperpolarizes. Thus, cell 1 and cell 2 have switched states after half the period: cell 2 is at the active state with V 2 at a maximal value and cell 1 is at the suppressed state with V 1 at a minimal value. The second half of the cycle is identical with the two cells roles switched. Fig. 1b plots the steady-state half-center oscillations projected onto a (V,n) 1

11 V i (mv) n i normalized phase θ / T (a) n i V i (mv) (b) Figure 1: Dynamics of a release-type HCO. η syn = 2 mv. (a) Top figure: membrane potentials V i of cell 1 (solid curve) and cell 2 (dashed curve) as functions of the phase (normalized by the period T ). The horizontal dotted line is the synaptic threshold η syn. Bottom figure: gating variables n i of cell 1 (solid curve) and cell 2 (dashed curve) as functions of the phase. The horizontal dotted line indicates the escape threshold n esc. In the release mechanism, the suppressed cell s gating variable is always above n esc. The transition between the active and the suppressed states is triggered by the active cell s membrane potential falling below η syn.(b)(v,n) phase plane. Solid curve: the V -nullcline when a cell is fully inhibited. Dashed curve: the V -nullcline when a cell is fully uninhibited. Dashed-dotted curve: the n-nullcline, which is not dependent on the status of the presynaptic cell. Filled circles: cell 1 s trajectory evenly spaced in time for the first half of the cycle. Open circles: cell 2 s trajectory evenly spaced in time for the first half of the cycle. The vertical dotted line indicates η syn. The horizontal dotted line indicates the escape threshold n esc, which is the 11 value of the gating variable n at the left knee of the fully inhibited V -nullcline.

12 phase plane. Because the two cells in the HCO model are identical and are firing in anti-phase, we can use the two-dimensional (V,n) phase plane to understand the dynamics of the 4-dimensional system. The evolution of the cells states, which correspond to the time series in Fig. 1a, are plotted as trajectories in the phase plane for the first half of the cycle. Points along the trajectories (solid dots for cell 1 and circles for cell 2) are plotted at equal time steps, thus the density of the points along the trajectory indicate the rates of change of the system. The n-nullcline (n = n (V ); dashed-dotted curve) is monotonically increasing in V. Because of the sharp threshold for synaptic activation, a cell is either inhibited (s (V pre ) = 1) or free (s (V pre ) = ), depending on whether the presynaptic cell s membrane potential is above or below η syn (plotted as a vertical dotted line). Accordingly, the V -nullcline for a cell is either the lower N-shaped curve (the inhibited V -nullcline; the solid curve) or the upper N-shaped curve (the free V -nullcline; the dashed curve). Because the HCO is assumed to be a relaxation-like oscillator, dynamics in the V -direction are much faster than those along the n-direction. This means that, if a cell is not near the V -nulllcine, the state of the cell will quickly jump to the corresponding stable branch of the V -nullcline (which are the branches with negative slopes in our example). At phase θ =, cell 1 is active and has a maximal membrane potential; cell 2 is suppressed and has a minimal membrane potential. As the cycle progresses, the state of cell 1 moves upwards in the phase plane closely adhering to the stable right branch of the free V -nullcline (i.e., slow hyperpolarization and increase of n 1 ). The dynamics of cell 1 slows down as it asymptotically approaches a stable fixed point for the isolated cells (V =13mV,n =.85). The state of cell 2 moves downwards in the phase plane along the stable left branch of the inhibited V -nullcline (i.e., slow depolarization and decrease of n 2 ). Before cell 1 reaches the fixed point, it crosses the synaptic threshold η syn from the 12

13 right shortly before phase.5t. This turns off the synaptic input from cell 1 to cell 2, leading to the release of cell 2 from inhibition. Upon its release, cell 2 s corresponding V -nullcline switches from the inhibited V -nullcline to the free V -nullcline, and the state of cell 2 jumps quickly to the stable right branch of the free V -nullcline (i.e., rapid depolarization of cell 2 with approximately constant n 1 ). During its jump, the membrane potential of cell 2 crosses η syn, leading to the inhibition of cell 1. Thus, cell 1 s dynamics switch to the inhibited V -nullcline, which leads to cell 1 s rapid jump to the stable left-branch of the inhibited V -nullcline (i.e., rapid hyperpolarization of cell 1 with approximately constant n 2 ). At phase θ =.5T, cell 1 is inhibited with its membrane potential at a minimum, and cell 2 is active (uninhibited) with its membrane potential at its maximal value. The dynamics for the second half of the cycle are identical to the first half with the roles of the two cells switched Escape Mechanism We now describe the dynamics of the HCO under the escape mechanism. Fig. 2a plots the membrane potentials V i and the gating variables for the outward K + current n i as functions of time for cell 1 and cell 2 during steady state oscillations. At phase θ =, cell 1 has just made a transition from the suppressed state to the active state. Its membrane potential V 1 is at a maximal value, which is above η syn, and therefore, cell 1 is inhibiting cell 2. Cell 2 has just been driven from the active state to the suppressed state by cell 1, and V 2 is at a minimal membrane potential. As the cycle progresses, the outward K + current in cell 2 slowly deactivates (i.e., n 2 slowly decreases), leading to a slow depolarization of cell 2. On the other hand, the outward K + current slowly activates in cell 1 (i.e., n 1 slowly increases), leading to a slow decrease in V 1.If it remain uninhibited, cell 1 would asymptotically approach an active steady state (V 1 =13mV,n 1 =.85). However, soon before phase θ =.5T (nor- 13

14 V i (mv) n i normalized phase θ / T (a) n i V i (mv) (b) Figure 2: Dynamics of an escape-type HCO. η syn = mv. Same conventions as in Fig. 1a and 1b. In the escape mechanism, the transition between the active and the suppressed states is triggered by the suppressed cell s gating variable reaching the escape threshold n esc. 14

15 malized phase.5), the gating variable n 2 for the outward K + current in cell 2 decreases to a critical value n esc =.13 below which the fast inward current is greater than the outward current. This leads to the rapid depolarization of cell 2, while it is still inhibited by cell 1. We refer to this critical value n esc as the escape threshold (indicated by the horizontal dotted line in the bottom figure in Fig. 2a). Upon its escape from inhibition, cell 2 s membrane potential V 2 rises above η syn, which turns on the synaptic inhibition from cell 2 to cell 1. The inhibition from cell 2 causes cell 1 to rapidly hyperpolarize, and V 2 falls below η syn, turning off the synaptic input from cell 1 to cell 2. Thus, at phase θ =.5T, cell 1 and cell 2 have switched states: cell 2 is in the active state with V 2 at a maximal value and cell 1 is in the suppressed state with V 1 at a minimal value. The second half of the cycle is identical to the first half of the cycle but roles of the two cells are switched. Fig. 2b plots the escape-type half-center oscillations described above in the (V,n) phase plane. The same notation and construction are used as in the phase plane described for the release case (Fig. 1b). At phase θ =, cell 1 is active and has a maximal membrane potential, and cell 2 is suppressed and has a minimal membrane potential. As the cycle progresses, the state of cell 1 moves upward in the phase plane adhering to the stable right branch of the free V -nullcline (i.e., slow hyperpolarization and increase in n 1 ), and it slows down as it approaches the stable fixed point of the isolated cell. Cell 2 s state moves downward along the stable left-branch of the inhibited V -nullcline (i.e., slow depolarization and decrease of n 2 ). Immediately before phase θ =.5T, cell 2 reaches the lower knee of the inhibited V -nullcline, which corresponds to the escape threshold n esc (plotted as a horizontal dotted line). Cell 2 is no longer restricted by the left stable branch of the inhibited V -nullcline, and it escapes by jumping rapidly towards the stable right branch of the inhibited 15

16 V -nullcline (i.e., rapid depolarization). During this jump, cell 2 s membrane potential crosses η syn, which turns on the synaptic inhibition from cell 2 to cell 1. Cell 1 s dynamics switch from the free V -nullcline to the inhibited V - nullcline, leading to cell 1 s rapid jump to the stable left branch of the inhibited V -nullcline (i.e., rapid hyperpolarization). Because cell 1 s membrane potential falls below η syn, cell 2 becomes uninhibited, leading to a switch of the inhibited V -nullcline to the free V -nullcline. As a result, cell 2 jumps further to the right onto the stable right branch of the free V -nullcline. At phase θ =.5T, cell 1 becomes the suppressed cell and cell 2 becomes the active cell. The dynamics for the second half of the cycle are identical to the first half cycle with the roles of the two cells switched. Note that, for the release-type HCO (Fig. 2), the gating variables n i for the slow outward K + current never fall below the escape threshold n esc. The cells transition between active and suppressed states does not rely on the intrinsic dynamics of the suppressed cell. Instead, the transition is triggered by the membrane potential of the active cell falling below the synaptic threshold η syn, which releases the suppressed cell from inhibition. On the other hand, for the escape-type HCO (Fig. 2), dynamics of the active cell and the exact value of η syn do not play a role in initiating the transition of the cells between active and suppressed states. The transition is controlled by the intrinsic properties of the suppressed cell. Specifically, as n i drops below n esc, the fast inward current overcome the slow inward current, leading to a rapid depolarization of the suppressed cell and its escape from inhibition. The phase portraits in Fig. 1b and 2b plot static nullclines for the fully inhibited (s = 1) and uninhibited (s = ) cases. When the full dynamics of the four-varaible HCO model is projected into (V,n) phase plane, the V i nullclines are continuously shifting because of their dependence on V j through the fast 16

17 synaptic input. Examining the dynamics of the system relative to the motion of the V i nullclines is helpful in understanding some of the subtleties of the half-center oscillations. Therefore, we provide movies of the dynamic (V,n) phase portraits for the release-type and escape-type half-center oscillations in the online supplemental material. 3.2 PRCs for Release and Escape Mechanisms Have Different Shapes Fig. 3a and 3b show the PRCs and corresponding membrane potentials for HCOs under the release and the escape mechanisms, respectively. Recall that because the two cells in the HCO are in anti-phase, the PRC obtained when cell 1 is perturbed (Z 1 (θ) =Z(θ)) and the PRC obtained when cell 2 is perturbed (Z 2 (θ) =Z(θ + T/2)) are identical except for a half-period shift. Therefore, we will describe only the PRC for cell 1, and refer to it as the PRC of the HCO. The PRCs for the two half-center oscillation mechanisms have strikingly different shapes. Most notably, the PRC for the release mechanism is dominated by a large negative peak late in the first half of the period, whereas the PRC for the escape mechanism is dominated by a large positive peak late in the second half of the period. This implies that a brief, positive applied current delivered to cell 1 (i.e., a perturbation to V 1 ) late in the first half of the period will significantly delay the phase of a release-type HCO, but it will do little to an escape-type HCO. On the other hand, the same perturbation delivered late in the second half of the period will significantly advance the phase of an escape-type HCO, but it will have a much smaller effect on a release-type HCO. Note that, despite their very different PRCs, HCOs under the release and escape mechanisms can have very similar membrane potential traces (see the bottom figures in Fig. 3a and 3b). This suggests that (1) it is difficult to predict a 17

18 Z i / T (1/mV) x phase advance phase delay Z i / T (1/mV) 5 x phase advance phase delay V i (mv) V i (mv) normalized phase θ / T normalized phase θ / T (a) (b) Figure 3: (a) ThePRCofarelease-typeHCO.η syn =2mV.(b)The PRC of an escape-type HCO. η syn = mv. Top figures: The PRCs of cell 1 (solid curve) and cell 2 (dashed curve). At phases when Z i >, the HCO will advance its phase in response to small, brief positive current pulses delivered to cell i; whereas when Z i <, the HCO will delay its phase in response to these stimuli. Bottom figures: The corresponding membrane potentials of cell 1 (solid curve) and cell 2 (dashed curve) as functions of the phase. The PRC of a release-type HCO is dominated by a large, negative peak late in the first half of the cycle, whereas the PRC of an escape-type HCO is dominated by a large, positive peak late in the second half of the cycle. 18

19 Z / T (1/mV) 15 x x x 1 3 Escape Release Release normalized phase θ / T normalized phase θ / T normalized phase θ / T Figure 4: PRCs of HCOs for different values of ɛ n and k syn. Left figure: η syn = mv, k syn =2. (allprcs);ɛ n =.2 (dashed curve), ɛ n =.5 (solid curve), ɛ n =.2 (dashed-dotted curve). Middle figure: η syn = 2 mv, ɛ n =.5 (all PRCs); k syn = 4 (dashed curve), k syn = 2 (solid curve), k syn =1 (dashed-dotted curve). Right figure: η syn =2mV(forallPRCs);ɛ n =.2 and k syn = 4 (dashed curve), ɛ n =.5 and k syn = 2 (solid curve), ɛ n =.2 and k syn = 1 (dashed-dotted curve). The left figure is based on the escape-type HCO, whereas the two figures on the right are based on the release-type HCO. The solid curve in the left figure is the same PRC showed in Fig. 3b. The solid curves in the two figures on the right are the same PRC showed in Fig. 3a. HCO s phase response properties simply by examining the membrane potential, and (2) it is difficult to assess whether the release or escape mechanism underlies the half-center oscillations from the membrane potential (without knowing the synaptic threshold and the escape threshold), but the oscillation mechanism is clear from the shape of the PRC. The basic shapes of PRCs for the release and escape mechanism persist as ɛ n (relaxation-oscillator-like dynamics) and k syn (sharp synaptic threshold) vary. The top figure in Fig. 4 shows the PRCs of escape-type HCOs with several values of ɛ n and fixed k syn =2.. The dominant positive peak late in the second half of the period is preserved, and in fact, it actually becomes taller and broader as ɛ n increases (i.e., as the dynamics of the system becomes less relaxation oscillator-like). Note also that the shape of the PRC for the escape-type HCO is insensitive to changes in k syn (not shown), because HCO is largely controlled 19

20 by intrinsic properties of the suppressed cell rather than the synaptic activation threshold (see Sec 3.1; Wang and Rinzel [45]; Skinner et al. [38]). The middle figure in Fig. 4 shows the PRCs of release-type HCOs with several values of k syn and a fixed ɛ n =.5. The dominant negative peak late in the first half of the period, which is characteristic of the PRCs for the release mechanism, is preserved. However, as the value of k syn increases (i.e., the synaptic threshold becomes less sharp), a small escape-like positive peak in the second half of the period increases in size. The bottom figure in Fig. 4 shows the PRCs of release-type HCOs when ɛ n and k syn are varied simultaneously. Once again, the basic shape of the PRC is unchanged, but larger values of both parameters combine to make the negative peak broader and less dominant. 3.3 How Escape and Release Dynamics Shape the HCO s PRC In this subsection, we explain how the dynamics of the HCOs give rise to its phase response properties. Specifically, we describe why a negative peak late in the first half of the cycle dominates the shapes of PRCs of release-type HCOs, and why a positive peak late in the second half of the cycle dominates the shapes of PRCs of escape-type HCOs. As before, our explanations use the idealizations of relaxation-oscillator-like dynamics (small ɛ n ) and a sharp synaptic threshold (small k syn ). However, we will show that the basic results hold when these restrictions are relaxed Responses of the Active Cell to Perturbations Dominates the PRCs of Release-Type HCOs Under the release mechanism, the cells transition between the active and suppressed states is controlled by the termination of the synaptic inhibition to the 2

21 Z / T (1/mV) normalized phase θ / T (a) Z / T (1/mV) 1 x normalized phase θ / T (b) Figure 5: (a) The dominant negative peak in the PRCs of release-type HCOs is largely determined by the response properties of the active cell. η syn =2mV. The PRCs of a release-type HCO with k syn =.2 (solid curve), k syn =.2 (dasheded curve) and k syn = 2 (dashed-dotted curve). Crosses respresent the normalized delays that result from small, brief positive perturbations delivered to an isolated cell. Specifically, an isolated cell is perturbed at different times as it relaxes towards its active steady state, and the changes in the time (delays) for the cell s membrane potential to reach η syn are measured. (Note that η syn is higher than the steady-state membrane potential). To make appropriate comparisons between the delays of the isolated cell with the PRC of the HCO, the delays are normalized by the stimulus strength (.1 mv) and the period of half-center oscillations. (b) The dominant positive peak in the PRCs of escape-type HCOs is largely determined by the phase response properties of the suppressed cell. η syn = mv. Solid curve: the PRC of an escape-type HCO. Crosses: The PRC of a cell that is constantly inhibited by the fully activated synapse. The positive peak in the PRCs of escape-type HCOs can be very well approximated by the positive peak in the PRC of the isolated suppressed cell. Recall that this escape-type HCO s phase response property is not sensitive to changes in k syn. 21

22 suppressed cell (see Sec. 3.1; [45, 6]). This suggests that the PRCs of releasetype HCOs can be understood by the effect of perturbations on the time at which the active cell s membrane potential crosses the synaptic threshold η syn. Indeed, Fig. 5a shows that the shape of the PRC for the release mechanism is largely inherited from the dynamics of an isolated uninhibited cell (i.e., the active cell) as it relaxes towards its active steady state. The crosses ( ) inthe figure represent the normalized delays that result from small, brief positive perturbations delivered to an isolated cell (see figure caption for details). The figure shows that, with a sufficiently sharp synaptic activation, the delays computed for the isolated uninhibited cell are indistinguishable from the negative peak in the HCO s PRC (solid curve). When η syn becomes larger, the synaptic threshold is not as distinct, therefore it is more difficult to compare quantitatively the responses of the isolated cell with the HCO s PRC. However, the negative peak in these smoother PRCs still results from the same basic mechanism. The relative flatness of the HCO s PRC in the second half of the period (i.e., the insensitivity to perturbations to the suppressed cell) can be explained as follows. Suppose cell 1 is initially the active cell and cell 2 in the suppressed cell. When cell 2 is perturbed, the perturbation will not affect the time at which cell 1 releases cell 2 from inhibition, but it could affect the time at which cell 2 releases cell 1 on the next half cycle. However, note that cells are relatively insensitive at the beginning of the active phase (as shown in Fig. 5a). Thus, even if the perturbation leads to a sizable effect on the state of cell 2 ((V 2,n 2 )) at the beginning of its active phase, the effect on the time at which cell 2 releases cell 1 will be relatively small. 22

23 3.3.2 Responses of the Suppressed Cell to Perturbations Dominates the PRCs of Escape-Type HCOs Under the escape mechanism, the cells transition between the active and suppressed states is controlled by the intrinsic dynamics of the suppressed cell. This suggests that the PRCs of escape-type HCOs can be understood by the effects of perturbations on the timing of the escape of the suppressed cell. Note that, in this case, if a cell were to be constantly inhibited (s = 1), it would undergo oscillations (see phase plane in Fig. 2b). Therefore, the effects of perturbations to the suppressed cell in the HCO can be characterized by the PRC of a cell that is constantly inhibited. Fig. 5b shows that the shape of the positive peak in the PRC computed for the constantly inhibited cell matches the shape of the dominant positive peak in the HCO s PRC. This implies that the shape of the second half of the HCO s PRC (i.e., the response due to perturbations to the suppressed cell) is largely inherited from the response properties the constantly inhibited cell. Under the escape mechanism, perturbations to the active cell have little effect, i.e., the HCO s PRC is almost flat in the first half of the cycle. In the case of escape-type HCOs, the active cell relaxes towards the active steady state and will not jump to the suppressed state unless the suppressed cell escapes. Therefore, any effect of perturbations to the active cell will exponentially decrease as the cell approaches this steady state, and any transient effect on the timing of the active cell will have no lasting effect on the phase of the full HCO. Moreover, cells are relatively insensitive at the beginning of the suppressed phase (as shown in Fig. 5b), therefore any small changes in state (timing) due to perturbations during the active phase will be further diminished. Note that we have linked the the shapes of the HCO s PRCs to the response properties of the individual cells. In the case of cells with relaxation-like dy- 23

24 namics, the mechanisms shaping individual cells s PRCs have been discussed by Coombes [4] and Izhikevitch [12]. 3.4 Half-Center Oscillations Involving Both Release and Escape Mechanisms In the previous subsections, half-center oscillations were described as being generated by either the release or escape mechanism. However, not every HCO can be classified as being distinctly release-type or escape-type. Dynamics of a HCO may involve elements from both the release and escape mechanisms, and therefore the PRC of a HCO may have both a negative peak late in the first half of the cycle and a positive peak late in the second half of the cycle. Nevertheless, the explanations presented in Sec 3.3 can still be used to understand shapes of PRCs for these less distinct cases. The PRCs of release-type HCOs in Fig. 4 (b) and (c) show that, when the synaptic activation curve is less steep (larger k syn ), a relatively small escapelike positive peak late in the second half of the period appears. This escape-like positive peak is in addition to the large negative peak late in the first half of the cycle that is characteristic of a release-type HCO s PRC. In the idealized case of the infinitely sharp synaptic threshold (k syn ), the synaptic inhibition was either on (s =1)oroff(s = ). In the case with a larger k syn, changes in the level of synaptic inhibition are smooth (η syn can no longer be thought of as an all-or-none threshold ). As the membrane potential of the active cell slowly decreases, the inhibition to the suppressed cell gradually weakens rather than instantaneously turns off. During this smooth release process, the value of the escape threshold gradually increases, and the suppressed cell reaches this escape threshold as it slowly depolarizes. Once the escape threshold is reached, the suppressed cell starts to depolarize rapidly to the active state. The 24

25 dynamics of the suppressed cell before its rapid depolarization are qualitatively similar to those described in the escape mechanism. Therefore, brief, positive perturbations delivered to the suppressed cell just before it escapes will have a significant phase advancing effect on the full HCO, and hence lead to the escape-like positive peak late in the second half of the cycle. In the phase plane, the change in the escape threshold during the smooth release process corresponds to the lower knee of the inhibited V -nullcline gradually shifting upward towards the free V -nullcline. Compared to the idealized case where the suppressed cell can not reach the left knee of its V -nullcline because the knee is too low, in the case where the synaptic activation is smoother, the left knee gradually rises during the release process and becomes within the reach of the suppressed cell, and hence leading to an escape-like dynamics late in the second half of the cycle. This gradual shift of the V -nullcline and the corresponding HCO dynamics are illustrated in the animated phase plane movies in the onlne supplemental material. Other parameters can also affect the oscillation mechanism underlying a HCO and hence the shape of its PRC. As an example, Fig. 6a depicts the PRCs of a HCO with a varying bias current in both cells in the HCO, I. Initially at I =.85 μa/cm 2, the PRC has a dominant negative peak late in the first half of period, which is the signature of the release mechanism. As I increases, the HCO and its PRC become less release-like and more escape-like. That is, the large negative peak becomes smaller, while a positive peak late in the second half of the period develops and grows larger. At an intermediate value I =.95 μa/cm 2, the PRC s positive and negative peaks are approximately equal in size, which implies that the two mechanisms play equally weighted roles in generating phase response properties. As I further increases to I =1.5 μa/cm 2,theHCO becomes distinctly escape-like with a dominant positive peak late in the second 25

26 x 1 3 I =.85 x 1 3 I =.9 x 1 3 I =.95 x 1 3 I = 1. x 1 3 I = Z / T θ / T (a).6.45 <Z> / T (1/mV) frequency (Hz) I (μa/cm 2 ) (b) Figure 6: (a) The PRCs of HCOs with different values of the bias current to both cells. As I (unit: μa/cm 2 ) increases, there is a smooth transition from a release-type PRC to an escape-type PRC. (b) Average of the PRC and frequency of oscillation as functions of I. Solid curve: the average of the PRC over a period, denoted by Z, normalized by T. When the PRC has a dominant negative peak, which is a signature of the release mechanism, Z is negative; whereas when the PRC has a dominant negative peak, which is a signature of the release mechanism, Z is positive. Dashed curve: the frequency of the half-center oscillations (f). f decreases when Z <, and increases when Z >. 26

27 half of the period. To illustrate that the transition from release to escape is smooth, the average of the PRC over a period ( Z = 1 T Z(θ)dθ) is plotted as a function of I in Fig. 6b (solid curve, normalized by the period T ). As I increases, Z smoothly increases from a negative value reflecting the dominant negative peak in release-type HCOs PRCs to a positive value reflecting the dominant positive peak in escape-type HCOs PRCs. T 3.5 Applications of PRCs Effects of Changes in Bias Currents on the Frequency of Half- Center Oscillations Fig. 6b shows that the frequency vs. bias current (f-i) curveofthehcois non-monotonic. When the HCO exhibits release-type behavior at lower values of I, f decreases with I; when the HCO exhibits escape-type behavior at higher values of I, f increases with I. The f-i curve of HCOs was studied in Shpiro et al. [37], Curtu et. al. [5] and Daun et al. [6]. The link between the HCO mechanism and the frequency dependence on the bias currents can be readily understood in terms of the PRCs. According to the 1 dφ phase model for the HCO (Eqn. 8 in Sec. 2.3), T dt, and therefore 1 T H(φ), is the instantaneous change in frequency Δf of an oscillator that results from the perturbation I ext. If the perturbation is taken to be a small constant change ΔI in the bias current in both cells of the HCO (i.e., I ext,i =ΔI, i =1, 2), then the change in HCO s frequency is Δf = 1 T H(φ) = 1 T 2 i=1 1 T T Z i (s + φ) ΔI 2ΔI ds = C CT Z. 27

28 Taking the limit ΔI, we see that df ( 2 ) di = Z. (9) CT That is, the instantaneous slope of the HCO s f-i curve is proportional to the average value of the PRC Z [34]. This relationship links the f-i curves, the PRCs and the oscillation mechanisms of HCOs. A release-type HCO has a negative average value of its PRC, which leads to the negative slope in the f-i curve; whereas an escape-type HCO has a positive average value of its PRC, which leads to a positive slope in the f-i curve The Phase-Locking Property of a Forced HCO Given a particular PRC Z(θ), the phase model (Eqn. 8) can be used to determine the phase-locking property of a HCO forced by a T -periodic external input. Here, we demonstrate that release-type HCOs and escape-type HCOs support very different phase-locking dynamics. Fig. 7a plots the functions H(φ) for both the release-type HCO and the escape-type HCO (as defined in Fig. 1a and Fig. 2a, respectively) when cell 1 is subject to a weak periodic inhibitory input. The input to cell 1 is I ext,1 (t) = ɛh ( (t mod T ).25T ),whereh is the Heaviside function and ɛ is the strength of the forcing. Recall that the zero crossings of H(φ) with negative slope correspond to the stable phase-locked states. We see that the release-type HCO is phase-locked to the periodic input I ext with a 19%-period advance relative to the stimulus, whereas the escape-type HCO is phase-locked to I ext with a 7%-period phase delay. Fig. 7b shows how the forced HCO s phase-locked states change as the bias current I changes. This figure is also known as the bifurcation diagram, which describes how the steady states of a dynamical system change as certain parameter changes. This figure is obtained by solving for the zero crossings of the H 28

29 1 H / ε normalized phase θ / T (a) 1 normalized phase θ / T I (μa/cm 2 ) (b) Figure 7: (a) The interaction functions H for two HCOs subjected to periodic forcing to cell 1. Solid curve: H function for the release-type HCO (from Fig. 1a). Dashed curve: H function for the escape-type HCO (from Fig. 2a). The arrow on the left points to the stable phase-locked state (at.19t )ofthe release-type HCO, and the arrow on the right points to the stable phase-locked state (at.93t or -.7T ) of the escape-type HCO. (b) Bifurcation diagram plotting the phase-locked states of periodically forced HCOs for different values of the bias current to both cells I. Solid curve: the stable phase-locked states for different values of I. Dashed curve: unstable phase-locked states at different values of I. Parameters are the same as those in (a) except that I is varied. 29

30 function for different values of I. The solid curve plots the stable phase-lockings (the zero crossing of H with a negative slope), and the dashed curve plots the unstable phase-lockings (the zero crossing with a positive slope). Starting from I =.6 μa/cm 2, the HCO originally has one stable phase-locked state around 2%-period and one unstable phase-locked state around 4%-period. As I increases, the stable phase-locked state remains around 2%-period until I reaches.975 μa/cm 2, at which moment the stable phase-locked state jumps to around 9%-period. For the unstable phase-locked state, as I increases it undergoes a saddle node bifurcation, however, since the saddle is very narrow in this case, the stable phase-locked state born in this saddle node bifurcation only persists for a very narrow range of I. It is important to note that I =.975 μa/cm 2 is a critical value here: when I<.975 μa/cm 2, the HCO is release-dominated (as illustrated in Fig. 6b), and a stable phase-locked state exists at around 2%- period; whereas when I>.975 μa/cm 2, the HCO is escape-dominated, and a stable phase-locked state exists at around 9%-period. This shows that different half-center oscillation mechanisms can lead to strikingly different phase-locked states for forced HCOs. 4 Discussion Phase response curve (PRC) describes an oscillator s pattern of phase shifts that are caused by inputs at different phases of oscillation. In neuroscience, PRCs are widely used to study the phase response properties of neural oscillators and predict synchronizations or phase-lockings in neural systems. Typically, each neural oscillator is viewed as an endogenous oscillator neuron, and hence, PRCs are usually computed for single cells. However, oscillations in neural systems often arise from network interactions, as in the case of half-center oscillators (HCOs). Despite this fact, there have been few theoretical studies that analyze 3