6.3.4 Action potential, Slides of Communication

Download 6.3.4 Action potential and more Slides Communication in PDF only on Docsity! 6 Mechnotransduction CmIion Cm dφ dt Figure 6.8: Electrical circuit model of the cell membrane. Normally, cells are net negative inside the cell which results in a non-zero resting mem- brane potential. The membrane potential of most cells is kept relatively stable. Nerve cells, skeletal, and cardiac muscle cells, however, are specialized to use changes in membrane potential for fast communication, primarily with other cells of their type. Within a millisecond, their membrane potential changes from positive to negative and back. This feature is referred to as action potential. 6.3.4 Action potential The action potential is a self-regenerating pulse-like wave of electro-chemical activ- ity that allows some cell types to rapidly carry signals over long distances. A typical action potential is initiated by a sudden change in the transmembrane potential. As the membrane potential is depolarized, both sodium and potassium channels begin to open generating an inward sodium current balanced by an outward potassium current. For only small perturbations, the potassium current wins and the membrane potential returns to its resting state. For sufficiently large perturbations of approximately 20 mV, however, the sodium current wins producing a positive feed back. The cell produces an action potential, we say the cell fires. One very important feature of the action po- tential is that its amplitude is independent of the degree of stimulation. Larger stimuli do not generate larger action potentials. This characteristic property of action poten- tials is referred to as all or none response, either the fires or it does not. The initiation and propagation of electrical signals by the controlled opening and clos- ing of ion channels is one of the most important cellular functions. Its first quantitative model was proposed half a century ago and awarded the Nobel Prize in 1963 [18]. Although originally developed for neurons, this theory was soon modified and gen- eralized to explain a wide variety of excitable cells. To gain a better understanding of these models, let’s take a look at equation (6.3.2) which we can rephrase as follows. φ̇ = − 1 Cm Iion with Iion = INa + IK + ICl + ICa2+ . (6.3.3) Here φ̇ = dφ / dt is the change in the transmembrane potential, Cm is the transmem- brane capacitance, and Iion is the total ionic current. This current results from the flux of sodium INa, potassium IK, chloride ICl, and calcium ICa2+ ions across the cell mem- brane. If we measured the transmembrane potential of different cells types found in the heart and plotted it over time, it would look somewhat like the illustraton in figure 6.9. Apparently, different cell types seem to have different action potentials. 77 6 Mechnotransduction sinus node atrial muscle AV node common bundle bundle branches Purkinje fibers ventri- cular muscle Figure 6.9: Electrophysiology of the heart: Characteristic action potentials and activation delay for vari- ous different cell types in the heart, adopted from [15]. So the key question in describing the curves in figure 6.9 with equation (6.3.3) is, what is the total ionic flux Iion that drives the evolution of the transmembrane potential φ? Two categories of models have been proposed to describe different action potentials: (i) ionic models and (ii) phenomenological models. Both are, of course, models and just a simplification of the reality. While ionic models represent the flux of all ions, the opening and closing of all channels, and the gating of ion channels in a very accu- rate way [4,25,32], phenomenological models actually only try to reproduce the action potential curve in a sufficiently accurate, but less expensive way [12, 22]. The FitzHugh-Nagumo model Probably the most prominent phenomenological model is the FitzHugh-Nagumo model [13,31]. It is based on an extremely elegant two-parameter formulation that allows the rigorous analysis of the underlying action potentials with well-established mathemat- ical tools. Most importantly, it allows for a graphic representation in the phase plane. Let’s look how the FitzHugh-Nagumo model can be derived. We start with a linear second order equation to describe the oscillations φ. φ̈ + k φ̇ + φ = 0 (6.3.4) In this equation, we replace the constant damping coefficient k with a quadratic term in terms of the potential k = c [ φ2 − 1 ] to obtain the following non-linear equation. φ̈ + c [ φ2 − 1 ] φ̇ + φ = 0 (6.3.5) With the help of Liénard’s transformation r = −1 c φ̇− 1 3 φ3 + φ ṙ = −1 c φ̈− [ φ2 − 1 ]φ̇ (6.3.6) this second order equation can be transformed into a system of two first order equa- tions. Its first equation follows from equation (6.3.6)1, its second equation follows from 78 6 Mechnotransduction tentials cannot follow one another immedaitely since the ion channels need to return to their resting state. The absolutely refractory period characterizes the period during which the cell is recovering. During this period, it is unable to generate a new action potential. Relatively refractory phase The relatively refractory phase is characterized through a decrease of the recovery variable r as the solution slowly returns to the resting state. During this phase, the ion channels gradually return to their initial state. A new action potential can be generated during this phase, however, the required stimulus might be significantly larger than in the resting state. Stable non-oscillatory and unstable oscillatory cells Action potentials occur when the cell membrane depolarizes and then repolarizes back to the steady state. There are two conceptually different action potentials in the heart: action potentials for pacemaker cells such as the sinoarial and the atrioventric- re co ve ry va ri ab le r potential φ po te nt ia lφ ,r ec ov er y r time t re co ve ry va ri ab le r potential φ po te nt ia lφ ,r ec ov er y r time t Figure 6.12: Phase portrait and physiological state diagram for stable non-oscillatory FitzHugh-Nagumo model with α = −0.1, top, and for unstable oscillatory FitzHugh-Nagumoe model with α = +0.1, bottom. For the non-oscillatory model, top row, the steady state, i.e., the intersection of nullclines, is situated to the left of the intersection in the stable regime. For the non-oscillatory model, bottom row, the steady state, i.e., the intersection of nullclines, is situated to the right of the intersection in the unstable regime. 81 6 Mechnotransduction ular node, and action potentials for non-pacemaker cells such as atrial or ventricular muscle cells. Pacemaker cells are capable of spontaneous action potential generation, whereas non-pacemaker cells have to be triggered by depolarizing currents from adja- cent cells. To compare non-oscillatory and oscillatory cells, it is convenient to rewrite the FitzHugh-Nagumo system (6.3.10) in a slightly modified form. φ̇ = c [ φ [φ + α][1− φ]− r ] ṙ = φ− b r− a (6.3.9) Based on this reformulation, we can easily distinguish between stable non-oscillatory muscle cells for α < 0 and unstable oscillatory pacemaker cells for α > 0, see figure 6.12. For the documented example, b = 0.5 and c = 100. Figures 6.12, top, show a stable non-osciallory pacemaker cell for α = −0.1. Right after the action potential, the membrane returns to its resting state. Figures 6.12, bottom, display a characteristic membrane potential for oscillatory cells for α = +0.1. The fast and slow variable undergo an oscillation through the four phase cycle of the regenerative, the active, the absolutely refractory, and the relatively refractory phase. After this cycle, however, the membrane potential is above the critical threshold to initiate a new excitation cycle. Traveling waves of excitation To account for the nature of traveling waves in excitable media, a phenomenologic diffusion term div(q) can be added to the first equation of the original FitzHugh- Nagumo equations. Based on the assumption that the spatial range of the signaling phenomenon φ is significantly larger than the influence domain of the recovery vari- able r, the second equation is considered to be strictly local. φ̇ = div(q) + c [−1 3 φ3 + φ− r + I ] ṙ = −1 c [ φ− b r− a ] (6.3.10) The easiest assumption is that the flux is proportional to the gradient of the membrane potential q = D ∇φ, where D denotes the conductivity. Typical conductivities in car- diac tissue are 0.05 m/s for the sinoatrial and the atrialventricular node, 1 m/s for atrial pathways, the bundle of his and ventricular muscle, and 4 m/s for Purkinje fibers [15]. Table 6.2 illustrates characteristic values for action potentials of different cell types. animal cell type resting potential potential conduc- potential increase duration tivity [mV] [mV] [ms] [m/s] squid (loligo) giant axon -60 120 0.75 35 earthworm (lumbricus) median giant fiber -70 100 1.00 30 cockroach (periplaneta) giant fiber -70 80–104 0.40 10 frog (rana) sciatic nerve axon -60 – -80 110–130 1.00 7-30 Table 6.2: Typical value of resting potential, action potential increase, action potential duration, and conduction speed for action potentials of different cell types. 82