Pacemaker Sensing Circuit

arXiv:physics/9905008v1 [physics.bio-ph] 5 May 1999

1

Limit Cycle Oscillations in Pacemaker Cells. Lars Petter Endresen and Nils Skarland∗. Abstract— In recent decades, several mathematical models de- scribing the pacemaker activity of the rabbit sinoatrial node have been developed.

Figure: System Model & Simulation Flow for Pacemaker Sensing Circuit

We Demonstrate That It Is Not Possible To Es-

tablish the existence, uniqueness, and stability of a limit cycle oscillation in those models. Instead we observe an infinite num- ber of limit cycles.

We Then Display Numerical Results From A

new model, with a limit cycle that can be reached from many different initial conditions. Keywords— Sinoatrial node, electrical activity, heart, mathe- matical model, nonlinear dynamics.

I. Introduction

In elementary electrostatics it is well known that the relation between the voltage and the charge of a capacitor is

(1)

where v is voltage, C is capasitance and q is charge. Differenti- ating this equation with respect to time we obtain

Dq

dt ≡i is a current, and the sign of v is a matter of convention. In physiology this second relation (2) is being used to describe how the membrane potential (v) is changed when ions move across the cell membrane. Unfortunately we observe that this equation also is being used in some models where one in addition keeps track of the charge (q) concentrations inside and outside the cell . In those models voltage and charge are believed to be independent dynamic variables: first one deter- mines the voltage by integrating the membrane currents, then one determines the charge by integrating the same membrane currents.

The purpose of this article is to point out that integrating the membrane currents once is enough. Voltage and charge cannot simultaneously be independent dynamical variables in a model, simply because of (1).

In order to visualize the drawbacks of treating voltage and charge as independent variables, we explore numerically the nonlinear dynamics of two different models describing the pace- maker activity of the rabbit sinoatrial node. The procedure is

As Follows:

1. we integrate numerically the equations of motion for a suffi-

Ciently Long Time To Detect A Steady State,

2. we change the initial conditions and repeat 1. First we display results from the Wilders et al. model , a model that treats voltage and charge as independent variables.

In that model it is thus possible to select an initial voltage and an initial charge independently. The dynamics of that model seems peculiar. An infinity of limit cycles is observed: each time we select new initial conditions a new limit cycle, corresponding to a new value of the constant of motion q −Cv, is found. This hampers the usefulness of the model.

Second we display results from a new model of Endresen et al. , where the voltage is not a dynamic variable. Here we cannot select an initial voltage independently of the initial charge, and only one limit cycle is observed.

Ii. Existing Models

The model of Wilders et al. of the pacemaker activity of the rabbit sinoatrial node serves as an excellent example of the many models where the membrane potential is thought to be independent of the intracellular and extracellular charge concen- trations. In that model the equation of motion for the voltage

Dt = −1

C (ib,Ca+ib,K+ib,Na+iCa,L+iCa,T+if+iK+iNa+iNaCa+iNaK) .

(3)

There are fifteen dynamic variables in that model, the voltage v, the gating variables dL, dT, fL, fT, x, y, h, m, p, and the ionic concentrations [Ca]i, [Ca]rel, [Ca]up, [K]i, [Na]i.

We Want To

determine how the long term dynamics in that model is changed when we change the initial conditions. To keep matters simple, we only change one initial condition: the initial intracellular concentration of potassium ([K]i); and we study the dynamics in two dimensions only: the phase space of v and [K]i.

Fig. 1

Phase portrait of the Wilders et al. model. In (a) the trajectories of the oscillator and in (b) the limit cycle and the initial conditions (P1) given by (4) and [K]i = 140.

Figure 1 displays the two–dimensional dynamics of the model

(4)

and [K]i = 140. In figure 1 (b) the point P1 denotes the v and [K]i coordinates of the initial conditions, and the closed loop at the bottom is the limit cycle. The trajectory of the model is displayed in 1 (a) where we observe that the model spirals

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