The Phase Response Curve of Reciprocally Inhibitory Model Neurons Exhibiting Anti-Phase Rhythms

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1 The Phase Response Curve of Reciprocally Inhibitory Model Neurons Exhibiting Anti-Phase Rhythms Jiawei Zhang Timothy J. Lewis Department of Mathematics, University of California, Davis Davis, CA 9566, USA September 3, 2 (Revised: June 8, 2) Abstract The phase response curve (PRC) of a half-center oscillator (HCO), which is formed by a pair of reciprocally inhibitory model neurons that exhibit anti-phase rhythms, is computed and used to identify the underlying mechanism of the HCO. The two distinct mechanisms, release and escape, give rise to two differently shaped PRCs. We use phase plane analysis to explain why the PRC is shaped differently under different mechanisms. Applying the theory of weakly coupled oscillators, we show that different mechanisms of a forced HCO can lead to different phase-lags. We also show that the distinction of the two differently shaped PRCs gets sharper as the system s time-scales diverge more and as the synaptic activation gets faster, and that when the system is away from its relaxation regime, it is common to observe a combination of both two mechanisms. Introduction A half-center oscillator (HCO) consists of two neurons coupled together through mutual inhibition. When decoupled, the two cells do not oscillate on their own; once coupled, they start to oscillate in anti-phase. The HCO can be modeled as a postinhibitory rebound (PIR) system using a minimal ionic model proposed by Wang and Rinzel [3]. Wang and Rinzel [3] showed that there are two different mechanisms underlying the HCO, release and escape, either of which is able to produce the anti-phase oscillation. It is difficult, if possible, to identify the HCO s mechanism based on the shape of the membrane potential of the two neurons as a function of time. However, we find that the different characteristics of the two mechanisms, release and escape, can be captured by the shape of their phase response curves (PRCs). (The PRC quantifies the phase shifts due to small delta-function inputs to an oscillator as a function of the phase of the input normalized to the amplitude of the input. We will give the definition of PRC in Section 2. For readers interested in more details about the PRC,

2 we refer them to []). Furthermore, because different mechanisms of the HCO lead to different phase response properties, the shape of the PRC may be used to study phase-lockings in a network of coupled oscillators. Our second paper will use the shape of the PRC, in conjunction with the theory of weakly coupled oscillators, to analyze phase-locking in a chain of HCOs underlying the crayfish swimmeret system. In this paper we illustrate the power of the PRC in revealing the mechanism of a HCO and in predicting the HCO s phase-lag property when subject to external synaptic inputs. In Section 2, we briefly review the minimal ionic model and how to compute the PRC. In Section 3 we show that two different shapes of the PRCs are observed under the two mechanisms, release and escape. In Section 4 we show that the distinction of the two PRCs gets sharper as the two time-scales in the system diverge more and as the synaptic activation gets faster. In Section 5 we use phase plane analysis to explain why the PRC is shaped differently under the two mechanisms. In Section 6 we show how the shape of the PRC changes as we adjust the maximum conductances of membrane currents and the existence of mechanisms that are a mixture of both release and escape. In Section 7 we use the theory of weakly-coupled oscillators to obtain the steady states of the phase-lags of a HCO subject to a weak periodic external input, and hence suggesting that different mechanisms of the HCO can lead to different phase-lockings in a network of coupled HCOs. We conclude this paper by a discussion in Section 8. 2 The PRC of a HCO In this section, we briefly describe Wang and Rinzel s minimal ionic model for a HCO and how to compute the PRC by using the adjoint of the original system. 2. The minimal ionic model We use the minimal ionic model proposed by Wang and Rinzel [3] for the dynamics of a HCO consisting of two neurons. Each neuron possesses just two intrinsic membrane currents: a constant conductance leakage current, I L, and a voltage-dependent inward current referred to as the PIR current, I PIR. Each neuron is assumed to be electrically compact and has two dynamic variables: membrane potential V and inactivation variable h for I PIR ; activation m is assumed instantaneous, i.e. m := m (V). The model equations for two identical cells coupled via inhibitory synapses are given by C dv i = m 3 (V i)h i ( Vi E PIR ) gl ( Vi E L ) gsyn s ji ( Vi E syn ), () dh i = φ ( ) h (V i ) h i, i=, 2, j=2,, (2) τ h (V i ) in which s ji is the postsynaptic conductance (fraction of the maximum g syn ) in cell i due to activity in cell j. We assume that s ji is an instantaneous sigmoid function of the presynaptic voltage with a 2

3 thresholdθ syn, i.e. s ji = S (V j )=/[+e (V j θ syn )/k syn ]. (3) Computations were done with k syn = 2 and g syn =.3 ms/cm 2. Reversal potentials have the values (in mv) E PIR = 2, E L = 6, and E syn = 8. The synaptic thresholdθ syn = 44 mv. With C = µf/cm 2 and g L =. ms/cm 2, the passive membrane time constantτ is msec. The factor φ scales the kinetics of h; φ = 3 unless stated otherwise. The voltage dependent gating functions are m (V)=/{+exp[ (V+ 65)/7.8]}, h (V)=/{+exp[(V+ 8)/]} and τ h (V)=h (V) exp[(v+ 62.3)/7.8]}. We use the same parameter values as Wang and Rinzel did [3] except for Section 6, where we study the dependence of the shape of the PRC on and g syn. Once the above parameters are fixed, the dynamic behavior of the two-cell system depends only on the remaining parameter: the maximum conductance of the PIR current,. For certain values of, the two neurons can form a HCO, i.e. the two neurons oscillate in anti-phase. 2.2 Compute the PRC by solving an adjoint problem For simplicity, let f (V i, V j, h i ) := [ m 3 (V i )h i ( Vi E PIR ) gl ( Vi E L ) gsyn s ji ( Vi E syn ) ]/C, g(v i, h i ) := h (V i ) h i, ɛ(v i ) := φ/τ h (V i ), i=, 2, j=2,. Then define f (V, V 2, h ) f (V F(X) := 2, V, h 2 ), ɛ(v )g(v, h ) ɛ(v 2 )g(v 2, h 2 ) where X=[V, V 2, h, h 2 ] T. To compute the PRC, we assume that the HCO has an asymptotically stable T-periodic limit cycle solution X LC. Then the PRC can be obtained by solving the following adjoint equation for Z(t) R 4 : dz = DF(X LC (t)) T Z (4) with the normalization condition Z() T F(X LC ())=, (5) in which DF is the Jacobian of F. Note that the first two components of Z(t) are the PRCs of the membrane potential of the two neurons, respectively. Furthermore, because the two cells in the HCO are identical and symmetrically coupled, the PRC of one neuron can be obtained by shifting the PRC of the other neuron by T/2. 3

4 The solution to (4) can be solved by integrating the equation backwards in time from an arbitrary initial condition, and using (5) to normalize the solution. This is because the original system has an asymptotically stable T-periodic limit cycle solution, which implies that the linearized adjoint system has an unstable T-periodic limit cycle solution. Note that the period of oscillation in every case is normalized, i.e. T=. 3 The PRCs for the Release and the Escape Mechanism Have Two Distinct Shapes Below we provide the plots of the membrane potential as a function of time of neuron V (t), the PRCs of the membrane potential of neuron Z (t), which is defined as the first component of Z(t), and the phase planes for the two mechanisms, release and escape, when the system is away from its relaxation regime (Figure 2). The values of the parameters used here are the same as those used in Wang and Rinzel s paper [3]. Memberane potential V (t) Phase plane V (mv) V 7 θ syn t (normalized) PRC of V h dv= (inhibited) dv= (free) dh= θ syn, "threshold" (V, h ) LC Z (t) t (normalized) V (mv) Figure : The release mechanism. =.3 ms/cm 2. As we can see, the PRC of the escape mechanism has a significant positive peak in the second half of the period. When it comes to the release mechanism, while the PRC has a significant negative peak in the first half of the period, it has a smaller positive bump in the second half, which suggests that the escape mechanism is involved, though not dominantly, in the release mechanism. This is due to the fact that the system is away from its relaxation regime. 4

5 Memberane potential V (t) Phase plane V (mv) V θ syn t (normalized) PRC of V h dv= (inhibited) dv= (free) dh= θ syn, "threshold" (V, h ) LC Z (t) t (normalized) V (mv) Figure 2: The escape mechanism. =.5 ms/cm 2. 4 Distinction of the Two Mechanisms Gets Sharper as the System Approaches Its Relaxation Regime In the release mechanism, the PRC has a small positive bump appearing after the tall negative peak, which is a characteristic of the escape mechanism. There are two reasons for this: One reason is that the fast time-scale is not infinitely fast. In Figure, suppose Neuron A is at V= 3 mv and Neuron B is at V= 7 mv initially. When Neuron B is released as Neuron A hitsθ syn, Neuron A has not yet jumped to the left only after Neuron B moves to the right ofθ syn will Neuron A quickly jump to the left side of the phase plane. Hence, a positive perturbation to Neuron B during its depolarization can bring forward the phase of the cycle. Another reason is due to the fact that S (V j ) is not perfectly steep; it takes time for the V nullcline to transform from the inhibited case (blue curve) to the free case (green curve). During this transition, the intersection of the h-nullcline and the V-nullcline moves from the left branch of the V-nullcline to the middle branch as the peak of the V-nullcline gets shorter. Hence, the escape mechanism is born during such a short period of transition. Therefore, in order to see a clearer picture of how the shape of the PRC depends on the mechanism, we shall bring the system closer to its relaxation regime so that we can deal with a purer mechanism. Two important parameters affect the distinction of the shapes of the PRCs: ()φ, which scales the ratio of the two time-scales in our system a smaller value results in more singular dynamics, i.e. the two time-scales deviate more; (2) k syn, which determines the steepness of the sigmoid function used for the postsynaptic conductance s ji = S (V j ) a smaller value results in faster activation of the synaptic coupling 5

6 current. Hence, by reducingφ, the peak on the PRC will look more like a delta function (see Figure 3); while by reducing k syn, the less likely there will be a combination of a positive peak and a negative peak (see Figure 4). Smaller values of these two parameters combine to create a sharper distinction between the release and the escape mechanism and more singular dynamics. PRC of V ("Escape", g pir =.5 ms/cm 2, θ syn = 44 mv) φ=3 φ=.5 φ=.3 Z (t) t (normalized) Figure 3: A smaller value ofφresults in more singular dynamics. Asφgets smaller, the positive peak the characteristic of the escape mechanism becomes thinner and taller, and the center of the peak approaches the phase when the cell starts to depolarize. The PRC of the release mechanism showed in Section 3 has a positive bump coming after the negative peak. By making the system more singular and the synaptic conductance steeper, we can flaten the positive bump (see Figure 5). Therefore, in the relaxation regime, the PRC of the release mechanism looks like a negative delta function centered at the phase that corresponds to the time when the membrane potential falls belowθ syn, the threshold of the synaptic current; while the PRC of the escape mechanism looks like a positive delta function centered at the phase that corresponds to the time when the neuron starts to depolarize. 5 Mechanism for Shaping the PRC in Singular Limit In this section we use phase plane analysis to explain why two different shapes of PRCs are observed for the two mechanisms. Here we assume that S is perfectly steep, i.e. the synaptic activation is instantaneous. 6

7 Z V (t) PRC of V ("Escape", g pir =. ms/cm 2, θ syn = 42 mv) 8 k syn =2 6 k =.8 syn k syn =.6 4 k syn =.4 k syn =.2 2 k =.2 syn t (normalized) Figure 4: A smaller value of k syn results in sharper distinction between the two mechanisms. As k syn gets smaller, the negative peak fades out, while the positive peak actually gets a bit thinner and taller. Equation 2 can be rewritten as dv i = f (V i, V j, h i ), (6) dh i =ɛ(v i )g(v, h i ). (7) We explore the fact that this system evolves roughly on two different time-scales. We assume thatɛ(v i ) ɛ, a constant. First consider the slow time-scale. If we letτ=ɛt and setɛ=, then Equation 6 7 is equivalent to = f (V i, V j, h i ), (8) dh i dτ = g(v,h i ). (9) This implies that on slow time-scale the trajectory crawls slowly upward or downward according to the sign of g on the curve f =. Note that depending on s ji, f = has two significantly different shapes; one corresponds the inhibited mode, the other corresponds to the free mode. The inhibited mode activates when the membrane potential of the other neuron rises aboveθ syn, and deactivates, or return to the free mode, when falls belowθ syn. 7

8 PRC of V ("Release", g pir =.3 ms/cm 2, θ syn = 44 mv) Z (t) φ=3, k syn =2 φ=.3, k syn = t (normalized) Figure 5: Smaller values ofφand k syn result in clearer release mechanism. (Simulated with =.3.) As these values get smaller, the dynamics of the system get more singular, the small positive peak fades out, and the negative peak the characteristic of the release mechanism gets thinner and taller. Now, consider the fast time-scale. If we setɛ=, then Equation 6 7 is equivalent to dv i = f (V i, V j, h i ), () dh i =. () This corresponds to the jump in either the positive or negative V direction depending on the sign of f on the fast time-scale. Below we explain the shapes of the PRC for the two distinct cases in detail. 5. The PRC under the release mechanism A simplified nullcline plot for the release case is showed below. Suppose that initially Neuron A is at () and Neuron B is at (3). As Neuron A slowly moves down from () to (2), Neuron B slowly moves up from (3) to (4). When Neuron A hits (2), Neuron B is instantaneously released and jumps to (). Hence, a perturbation to Neuron A in the positive (negative) direction of V-axis near (2) will more significantly delay (advance) the phase of the cycle than at a location away from (2). This implies that on the PRC there should be a negative peak at the time when the neuron is at (2). Note that ifθ syn < V, where V is the potential at the intersection of the curves dv= (free) and dh=, then the system will reach a bistable state because Neuron B will not be released as the potential of Neuron A can not go down toθ syn. 8

9 Phase plane.9.8 h dv= (inhibited) dv= (free) dh= θ syn, "threshold" (4) (). (3) (2) V (mv) Figure 6: The release mechanism. 5.2 The PRC under the escape mechanism A simplified nullcline plot for the escape case is showed below. Suppose that initially Neuron A is at () and Neuron B is at (3). As Neuron A slowly moves down from () to (2), Neuron B slowly moves up from (3) to (4). When Neuron B hits the peak of the curve dv= (inhibit) at (4), Neuron B immediately jumps towards the right branch of the curve dv= (inhibit). Furthermore, Neuron B crossesθ syn before it arrives at the right branch, which releases Neuron A. As Neuron A jumps to the left branch of dv= (inhibit), Neuron A crossesθ syn, which brings Neuron B to () on the curve dv= (free). In comparison to the release mechanism, Neuron B will escape from the left branch of dv= (inhibit) on its own regardless of whether the membrane potential of Neuron A falls belowθ syn. Hence, a perturbation to Neuron A near (2) almost has no effect on the phase as (2) is a stable node, instead of an intersection of the curve dv= (free) and the line V=θ syn in the release mechanism. On the other hand, a perturbation to Neuron B near (4) in the positive (negative) direction of V- axis will advance (delay) the phase because a positive perturbation will push Neuron B into the fast time-scale and thus activate the escape, while a negative one will delay this process. This implies that on the PRC there should be a positive peak near the time when the neuron is at (4). 9

10 .3.25 Phase plane dv= (inhibited) dv= (free) dh= θ syn, "threshold".2 h.5 (4) ()..5 (3) (2) 5 5 V (mv) Figure 7: The escape mechanism. 6 Dependence of the PRC s Shape on and g syn In the ideal relaxation regime, i.e.ɛ is very close to and S is very close to a step function, we can not see a combination of the two mechanisms because one of the two neurons is either released by the other neuron or escapes on its own. However, when the system is away from its relaxation regime, we usually have a mechanism that involves both release and escape. Figure 8 shows that the shape of the PRC changes as we adjust the maximum conductance of the PIR current ( ) and that of the postsynaptic current (g syn ). As we can see, not every case can be clearly identified as either release or escape; it is usually a mixture of both mechanisms. In most cases, a bigger value of gives a mechanism dominated more by escape; a bigger value of g syn gives a purer mechanism, i.e. either the positive peak in the second half of the period dominates ( escape ) or the negative peak in the first half period dominates ( release ). Note that two of nine cases presented in Figure 8 are near bifurcation: =.5, g syn =. and =.9, g syn =.3.

11 =.3, g syn = =.9, g syn =. 2 4 =.5, g syn =..5.5 =.3, g syn =.2 5 =.9, g syn =.2 5 =.5, g syn =.2 =.3, g syn = =.9, g syn =.3 2 =.5, g syn = Figure 8: Dependence of the PRC s Shape on and g syn. (Forgot to add labels for the axes!!!) 7 A Forced HCO: Different Mechanisms Lead to Different Phase- Lags In this section we consider how a weak periodic external excitatory synaptic current (See Figure 9) affects the phase-lag of the HCO under the two different mechanisms: The escape ( =.5 ms/cm 2 ) and the release ( =.3 ms/cm 2 ). Figure 9: Cell of the HCO is receiving an excitatory synaptic current from an external source. We apply the theory of weakly-coupled oscillators to compute the phase-lagged steady states of the HCO. Specifically, we are looking for the roots of the H function, which is defined in the Appendix. Solving H(φ )= forφ gives all the phase-lagged steady states. φ is a stable

12 (unstable) phase-lagged steady state if H (φ )< (> ). (See Figure.) (Note: Theφhere is not the parameterφdefined in the minimal ionic model in Section 2.) Numerical results are listed below in Table. Both of the two mechanisms lead to two unique phase-lagged steady states, one stable and one unstable. However, the two mechanisms give significantly different values for the stable state. We can see from Figure that, if subject to noises, the release mechanism would give a more stable phase-lag. Table : Phase-lagged steady states of a HCO subject to a weak periodic external excitatory synaptic current Mechanism Stable phase-lagged state Unstable phase-lagged state Escape.9.25 Release H function H(φ) Escape Release Phase lag φ (normalized over the period) Figure : The H functions for the two mechanisms. 2

13 8 Discussion We have showed that the two distinct mechanisms, release and escape, underlying the antiphase oscillation exhibited in a postinhibitory rebound (PIR) system give rise to two different shapes of the phase response curve (PRC) of the half-center oscillator (HCO) formed by this PIR system. Since PRCs can be obtained from experiments, we can use them to determine the underlying mechanisms of HCOs. We showed that the distinction of the two shapes of the PRC gets sharper as the system gets more singular, i.e. the time-scales diverge more, and as the synaptic activation gets faster. By using phase plane analysis, we explained the reason why the two different mechanisms give rise to two different shapes of the PRC. We also showed how the shape of the PRC varies as we adjust the maximum conductance of the PIR current and that of the postsynaptic current. When the system is away from its relaxation regime, we usually have a mechanism that involves both mechanisms. Finally we showed that different mechanisms of the HCO can lead to different phase-lags when the HCO is subject to external synaptic inputs. This suggests that the shape of the PRC may be used to understand phase-lockings in coupled oscillator networks. When the system is not close to its relaxation regime, it is easy to find an almost pure escape mechanism (e.g. Figure 2), but not easy to find an almost pure release mechanism. This is because in the release case when the inhibited cell is being released as the free cell s membrane potential falls below the threshold of the synaptic current, the peak of the V-nullcline moves down under the h-nullcline, and hence giving birth to the escape mechanism, which introduces a positive bump following the negative peak on the PRC. Skinner, et al [2] suggested that there are at least four different mechanisms in a network of two reciprocally inhibitory cells based on the Morris-Lecar model. We think that the four mechanisms still only give rise to two different shapes of PRCs. Hence, for the phase-locking property of coupled oscillators, we only need to consider the two mechanisms, release and escape, proposed by Wang and Rinzel [3]. Appendix Here is our model for the HCO subject to a T periodic external inputting current: I C ext(t) dx = F(X), (2) where X=[V, V 2, h, h 2 ] is a column vector that consists of the four variables associated with the the HCO. Note that the subscripts and 2 denote the two neurons inside the HCO. In order to obtain the phase-lag property of the the HCO, we apply the theory of weakly coupled oscillators and averaging theory to the above model. Readers unfamiliar with the theory may want to read [] for more details. Assume the HCO when isolated, i.e. I ext =, has an asymptotically stable T-periodic limit cycle solution denoted by X LC (t). We can find a mappingφto take the variables X(t) to their 3

14 corresponding phase, i.e. Then we have dθ θ(t)=φ(x(t)). (3) = X Φ(X(t)) dx(t) = X Φ(X(t)) [ F(X) I C ext(t) = X Φ(X(t)) C I ext(t) ] (4) (5). (6) If the external input is weak, the intrinsic dynamics of the the HCO dominate the dynamics of the perturbed system, which implies X(t) X LC (t+φ)=x LC (θ), (7) whereφ :=θ t is the relative phase of the HCO. In the following we assume that X(t)=X LC (t+φ). For convenience, denote by V LC the first component of X LC, the membrane potential of the neuron. Then we have V (t)=v LC (t+φ). Because the two cells in the local HCO are in anti-phase, we have V 2 (t)=v LC (t+φ+t/2), where T is the period of the system. We have already computed the PRC Z(t) of the local HCO system dx = F(X) by solving an adjoint problem. Note that X Φ(X LC (t))=z(t) and take the time derivative ofφ, we obtain an ODE: dφ = Z(t+φ) C I ext(t). (8) Because changes in the relative phaseφoccur on a much slower time scale than the period of the intrinsic oscillation T, we can apply averaging theory and obtain an ODE whose solution approximates the original solution with an error in the order of O(ɛ 2 ), whereɛ is the ratio of the fast time scale T f ast over the slow time scale T slow (roughly speaking). We obtain our phase model: dφ = T Z ( t)i ext ( t φ)d t (9) CT =: H(φ), (2) where Z (t) is the first component of the vector Z(t) (Z (t) is the PRC of V ). φ is the phase-lag of the HCO. We define the right-hand-side of Equation 9 by H(φ). Solving H(φ )= forφ gives all the steady states of the phase-lags.φ is a stable (unstable) phase-lagged steady state if H (φ )<(> ). 4

15 References [] Michael Schwemmer and Timothy J. Lewis, The theory of weakly coupled oscillators, PRCs in Neuroscience: Theory, Experiment and Analysis, Springer (to appear). [2] Francis K. Skinner, N. Kopell and E. Marder, Mechanisms for oscillation and frequency control in reciprocally inhibitory model neural networks, J. Comput. Neurosci. :69 87 (994). [3] Xiaojing Wang and John Rinzel, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neuron Comput. 4:84 97 (992). 5