MATH 723 Mathematical Neuroscience Spring 28 Instructor: Georgi Medvedev 2 Lecture 2. Approximate systems. 2. Reduction of the HH model to a 2D system. The original HH system consists of 4 differential equations. In the present lecture, we introduce several approximations which allow to reduce the HH model to a system of 2 differential equations. The analysis of the approximate 2D model provides an insight into the mechanism for spike generation in the HH model. The following oservations lead to the desired reduction: Step. Note that τ m (V ) τ h (V ),τ n (V ) (see Figure.2),i.e., the dynamics of the activation of the sodium current has the fastest time scale. Therefore, to leading order, we approximate m(t) y its steady state value: m m (V ). The resulting system has the following form: V = g Na m 3 (V )h(v E Na ) g K n 4 (V E K ) g L (V E L ) + I, (2.) ṅ = n (V ) n, (2.2) τ n (V ) ḣ = h (V ) h. (2.3) τ h (V ) Step 2. The following oservation is due to Krinsky and Kokoz [KK]. The dynamics of h is similar to that of n. The numerics in Figure 2. show after a short period of time necessary for transients to settle down, h + n. More precisely,.n + h.89 (see Figure 2.). By approximating h l(n), l(n) =.89.n, we arrive at the following system of two differential equations: V = g Na m 3 (V )l(n)(v E Na ) g K n 4 (V E K ) g L (V E L ) + I, (2.4) ṅ = n (V ) n, (2.5) τ n (V ) Step 3. Finally, it is convenient to approximate steady-state functions m (v) and n (v) and time constant τ n (v) using hyperolic functions [BRS]: { ( )} V θs s (V ) = + exp, s = m,n, (2.6) σ s ( ) V θn τ n (V ) = τ n cosh, (2.7) 2σ n The original functions and their approximations (2.6) and (2.7) are shown in Figure 2.2a. The parameter values for the reduced model (2.4) and (2.5) are summarized in
V (mv) 2 8 6 4 2 V(t) 2 2 3 4 5 6 n(t) + h(t) h.8.6.4 h =.89.n.5 h(t).2 a n(t) 2 3 4 5 6 t (ms).2.4.6.8 n Figure 2.: Plots in a and show that variales n and h are approximately linearly dependent. Tale 2. g Na 2mS/cm 2 E Na 2mV θ m 25mV σ m 9.5 g K 36mS/cm 2 E K 2mV θ n.3mv σ n 9.5 τ n 2 g L.3mS/cm 2 E L.6mV C µf/cm 2 2.2 The rescaled system. Variales v and n in the reduced HH model have different models of magnitude. During an AP, voltage varies from 2mV to aout mv, while the values of the gating variale range from.4 to. To analyze (2.4) and (2.5), it is convenient to rescale the dynamical variales so that they have the same orders of magnitude: V = k V v, k V = mv, τ = k t t, k t = ms. (2.8) We also rescale the maximal conductances and the reversal potentials: g s = g s mscm 2 and E s = Ẽs k v, s = {m,n}. Thus, we otain the following nondimensional system: ǫ v = g Na m 3 (v)h(v ẼNa) g K n 4 (v ẼK) g L (v ẼL) + Ĩ,, (2.9) ṅ = ñ (v) n. (2.) τ n (v) where s (v) = s (k V v), s = {m,n} and τ n (v) = τ n (k V v). The derivative of v is multiplied y a small nondimensional parameter ǫ = 2. This indicates that that the dynamics of v and n have different timescales. Equations (2.) and (2.) imply that typically: v = O ( ǫ ) while ṅ = O(), i.e., v changes 2
.8 m (v).8 n (v) 6 5.6.4.6.4 4 3 2 τ n (v).2.2 a 5 V (mv) 5 V (mv) 5 5 V (mv).4.2.8.6.4.2 -.2 5 5 2 25 3 35 4 Figure 2.2: a. Steady-state functions m (v) and n (c), and and time constant τ n (v) (solid line) and their approximations y the hyperolic functions (dashed line).. Action potential generated y the 2D approximation of the HH model y applying a step current pulse. 3
a.8.8.6.6.4.2.4.2 a -.4 -.2.2.4.6.8.2.4 -.2 2 4 6 8 2 4 6 8.8.8.6.6.4.2.4.2 -.4 -.2.2.4.6.8.2.4 -.2 2 4 6 8 2 4 6 8 Figure 2.3: The rescaled system: the phase plane and time series: a) excitale regime and ) oscillating regime. much faster than n. For this reason, v is called a fast variale. The v time series in Figure 2.3 show alternating intervals of fast and slow dynamics. Systems of ODEs with small parameters multiplying timederivatives in some of the equations are called singularly pertured or fast-slow systems. In the following susection, we descrie a geometric method for analyzing a 2D singularly pertured system. 2.3 The FitzHugh-Nagumo model: slow-decomposition decomposition Below we outline some ideas used for analyzing singularly pertured systems of ODEs. To make our discussion more explicit, we will use a specific example of a 2D singularly pertured system, the FitzHugh- Nagumo model: ǫ v = f(v) n + I, (2.) ṅ = v γn, (2.2) where v and n are the dynamical variales, and I,γ >, < ǫ are parameters. A smooth function f(v) is qualitatively cuic, i.e. it has two extrema and lim v ± f(v) = ±. A small parameter ǫ > is called a 4
5 4 N v N n L v 2 (v o, n o ) n 5 2 5 (v * (n o ), n o ) 5 a c d n 5 a 5 2 2 4 v 5 Figure 2.4: The FitzHugh-Nagumo model: a. The phase plane,. Solution plots. relaxation parameter. Along with (2.)-(2.2), we shall consider the following system of equations: dv dτ dn dτ = f(v) n + I, (2.3) = ǫ(v γn), (2.4) where τ = ǫ t is called the fast time. By formally taking the limit as ǫ in (2.3) and (2.4) we arrive at the fast susystem: dṽ dτ dñ dτ = f(ṽ) ñ + I, (2.5) =. (2.6) Let (v,n ) e the initial condition for (2.5), (2.6), as shown in Fig 2.3. For almost all initial conditions (v,n ), v elongs to the domain of attraction of one of the stale fixed points of the D ODE: dṽ dτ = f(ṽ) n + I. Denote this fixed point y v (n ). The trajectory of the fast system (2.5), (2.6) starting at (v,n ) tends to (v (n ),n ) as τ along the line n = n (see Figure 2.4). The vector field generated y (2.5),(2.6) is called the fast vector field. The theory for singularly pertured systems implies that for sufficiently small ǫ >, the solutions of the fast system (2.5) and (2.6) approximate those of the original system (2.3) and (2.4) with the (same initial condition) up to the time when the trajectory of the fast system enters some O(ǫ) neighorhood of (v (n ),n ). To study the dynamics of (2.3) and (2.4) after that moment of time, we consider original system (2.), and (2.2). Again, y formally taking the limit as ǫ, from (2.3) and (2.4) we derive the slow system: n = f(v) + I, (2.7) ṅ = v γn, (2.8) 5
The dynamics of the slow system is constrained y the two outer (stale) ranches of the slow curve n = f(v) + I. The set of points with coordinates satisfying (2.7) is also called the v nullcline: N = {(v,n) : n = f(v) + I} (see Figure 2.4). The motion along N is determined y the equation (2.8). By differentiating oth sides of (2.7) with respect to time and y comining the result with (2.8), we otain: By plugging in (2.7) into (2.9) we otain dv dt f (v) dv dt = v γn, (2.9) v γ(f(v) + I) = f, (2.2) (v) Equation (2.2) is valid outside of some neighorhoods of the points of extrema of f(v) where f (v) =. According to (2.2) on the slow manifold trajectory with initial condition, as shown in Figure 2.4, will move down along the left hand outer ranch of N v until it reaches an O(ǫ) neighorhood of a. After that the point will jump to the opposite stale ranch of N v under the action of the fast vector field. The description of the slow dynamics along this ranch is completely analogous to that for the left outer ranch. Therefore, after some finite time the phase trajectory remains in the neighorhood of the closed curve L composed of the segments of the slow manifold joined y horizontal segments (see Figure 2.4a). L is called a relaxation limit cycle. The alternation of the slow motions and fast jumps generates relaxation oscillations (see Figure 2.4). The aove discussion shows that for a 2D singularly pertured systems such as (2.3) and (2.4), the phase plane is to a large extent is determined y the geometries of the v and n nullclines: n = f(v) + I and n = γ v. In the FHN model, parameter I plays the same role as the applied current in the HH model. Let us consider how the variation of I affects the qualitative dynamics of the FHN model. Note that changing I results in the translation of the voltage nullcline in the vertical direction. In particular, y decreasing I we can achieve that the point of the intersection of the two nullclines ( v, n) lies on the left outer ranch of L (see Figure 2.4). Note that at (v,n) = ( v, n) the right-hand side of (2.) and (2.2) are zero. Therefore, ( v, n) is a steady state or a fixed point. For the FHN, if the fixed point is located on the stale ranch of the slow curve it attracts the trajectories following it. Therefore, after a finite time the dynamics of the system approaches the steady state ( v, n). The phase plane discussion for the FHN system gives us a first insight into the mechanism of the excitation in the HH system. By changing the applied current in the HH system, we change the character of the fixed (it ceases to e attracting) and create an attracting limit cycle similar to L in the present example. 2.4 Two types of neuronal excitaility We conclude this lecture with an introductory discussion of the two major mechanisms for neuronal excitaility (cf. [RE]). For this, we will use the phase plane techniques outlined in the present section. In 6
a.8.8.6.6.4.4.2.2 c - -5 5 d - -5 5.8.8.6.6.4.4.2.2 - -5 5 - -5 5 Figure 2.5: Phase portraits plotted for the ML model with different parameter values, illustrating two mechanisms of excitaility: (a,) Type II, (c,d) Type I. the previous susection, we showed that y changing the value of the applied current, I, one can affect the position of the fixed point in the FH system. The fixed point moves from the left ranch to the middle ranch of the cuic nullcline under the increase in I. For low values of I, the fixed point lies on the left ranch and it locks the motion of the trajectories along the slow manifold. Shortly after the fixed point passes the lower fold of N v under the variation of I, it opens the way for the trajectories starting near the left ranch to jump toward the right ranch and gives rise to a limit cycle. This is an example of an excitale system, i.e., a system where a significant deviation from a stale steady state can e induced y a relatively small change of a control parameter. A similar mechanism is responsile for generation of the AP in the reduced HH system (Fig. 2.5a,). Next we discuss a different excitaility mechanism illustrated in Fig. 2.5c,d. As in the previous case, the system shown in Fig. 2.6c is in an excitale state: the fixed point is lying on the left ranch of the cuic nullcline. However, in contrast to the case discussed aove, under the increase of applied current the fixed point collides with another one and disappears (Fig. 2.5d). As a result the limit cycle is orn. The representative time series corresponding the two scenarios of transition to spiking are shown in Fig. 2.7 and 2.7. Note that in Fig. 2.6, the frequency of oscillations just near the transition is extremely low. This turns out to e a distinctive feature of a road class of models, Type I models. The FHN model and the 7
a 4.8 2.6.4-2.2-4 -6 - -5 5 5 5 2 25 3 3 2.8.6 -.4-2.2-3 -4-5 c - -5 5 5 5 2 25 3 4.8 2.6.4-2.2-4 -6 - -5 5 5 5 2 25 3 Figure 2.6: Phase portraits and the time series illustrating transition to spiking in Type I models. reduced HH model elong to the Type II class of models. The frequency of oscillations is ounded from zero near the transition to spiking in Type II models. Explanation of these effects e given later after a rief introduction to elementary ifurcation theory in Lecture 4. Homework project II. a. Plot the phase portrait in the ML model. Determine whether it is a type I or type II model. Change the value(s) of some parameter(s) in this model to make it of a different type.. For the ML model in the Type II regime, use the phase plane analysis (along the lines of Section 2.3) to descrie the mechanism of the AP generation under electrical stimulation. Derive the formula for the period of the relaxation limit cycle. Biliography BRS R.J. Butera, J. Rinzel, J.C. Smith, Models of Respiratory rhythm generation in the Pre-Botzinger 8
a 3 2.8.6 -.4-2 -3.2-4 -5 - -5 5-6 5 5 2 25 3 3 2.8.6.4 - -2.2-3 -4 c - -5 5 5 5 2 25 3 4.8 2.6.4-2.2-4 - -5 5-6 5 5 2 25 3 Figure 2.7: Phase portraits and the time series illustrating transition to spiking in Type II models. 9
Complex: I. Bursting pacemaker neurons, J. of Neurophys. 8: 382-397, 999) KK VI Krinsky and YM Kokoz, Analysis of equations of excitale memranes I. Reduction of the Hodgkin- Huxley equations to a second order system. Biofizika, 8, 56 5, 973. MKKR E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov, Asymptotic Methods in Singularly pertured Systems, Consultants Bureau, New York, 994. RE J. Rinzel and G.B. Ermentrout, Analysis of neural excitaility and oscillations, in C. Koch and I. Segev, eds Methods in Neuronal Modeling, MIT Press, Camridge, MA, 989.