HODGKIN-HUXLEY MODEL: MEMBRANE CURRENT

Transcription

1 DOING PHYSICS WITH MATLAB BIOPHYSICS HODGKIN-HUXLEY MODEL: MEMBRANE CURRENT Ian Cooper School of Physics, University of Sydney MATLAB SCRIPTS (download files) alpha. Function: calculates the rate constants x (equations 18, 19) beta. Function: calculates the rate constants x (equations 18, 19) alpha_beta_plot. -script: plot of rate constants as a function of ebrane voltage (figure 4) N_0. Function: calculations n(0), (0), h(0) (equation 13) N_inf. Function: calculations n(), (), h() (equation 13) tau. Function: calculations of tie constants n,, h (equation 15) tau_nh_inf_plot. -script: plots of n(), (), h() and n,, h (figure 5) bp_neuron_02. -script: plot of rate constants x, x and gate variables gk, gna (figure 7) for a voltage clap bp_neuron_01. -script: plots of ebrane voltage and conductances for an external current stiulus bp_neuron_01a. -script: plots of ebrane voltage and conductances for an external current stiulus of ultiple pulses bp_neuron_01b. -script: plots of ebrane voltage and conductances for an external current stiulus bp_neuron_01bb. -script: plot of firing frequency vs agnitude of current stiulus (figure 16) bp_neuron_01c. -script: different external current stiuli (can include noise) and plot of interval distribution for repetitive firing Doing Physics With Matlab bp_hh_01 1

2 Physical Quantities: Sybols, Units, Typical values t ~ s tie a 5x10-6 Radius of axon b 6x10-9 Mebrane thickness R i Longitudinal resistance of axon r in 6.4x Longitudinal resistance / length axoplas resistance / length V(t,x) V (t,x) ~ - 70 V to + 40 V instantaneous potential difference across ebrane (ebrane potential) V in (t,x) ~ - 70 V to + 40 V potential inside cell Vin = V (ebrane potential) V out 0 potential outside cell: consider as ground potential V R -65 V Resting ebrane potential C F ebrane capacitance c 1.0x10-6 F/c 2 Capacitance / area Q(t,x) C instantaneous charge on capacitor plates I C (t,x) A instantaneous capacitive current I ion (t,x) = A ion current or ebrane current I (t,x) I Na (t,x) A Na + current I K (t,x) A K + current I L (t,x) A leakage current sall, ainly Cl - I in (t,x) A Longitudinal current current along inside of axon J ext (t,x) A.c -2 Instantaneous external stiulus current density J ion (t,x) = A.c -2 ion current or ebrane current density J (t,x) J Na (t,x) A.c -2 Na + current density J K (t,x) A.c -2 K + current density J L (t,x) A.c -2 leakage current density sall, ainly Cl - J ext (t,x) A.c -2 external stiulus current density G Na (t,x) -1 Na + conductance G K (t,x) -1 K + conductance G L (t,x) -1 leakage conductance g Na (t,x) g Na = 120x c -2 Na + conductance / area axiu value g K (t,x) g K = 36x c -2 K + conductance / area axiu value g L (t,x) g L = 0.3x c -2 leakage conductance / area axiu value E Na + 50 V reversal potential for Na + E K - 77 V reversal potential for K + E L V reversal potential for leakage Doing Physics With Matlab bp_hh_01 2

3 k 1.38x10-23 J.K -1 Boltzann s constant e 1.60x10-19 C Charge on an electron z Valency of an ion T K or o C Teperature Doing Physics With Matlab bp_hh_01 3

4 INTRODUCTION The core atheatical fraework for odern biophysically based neural odelling was developed around 1950 by Alan Hodgkin and Andrew Huxley. They carried out a series of elegant electrophysiological experients on the squid giant which has an extraordinarily large diaeter ~ 0.5. Hodgkin and Huxley systeatically deonstrated how the acroscopic ionic currents in the squid giant axon could be understood in ters of changes in Na + and K + conductances in the axon ebrane. Based on a series of voltage-clap experients, they developed a detailed atheatical odel of the voltage-dependent and tie-dependent properties of the Na + and K + conductances. Their odel accurately reproduces the key biophysical properties of the action potential. For this outstanding achieveent, Hodgkin and Huxley were awarded the 1963 Nobel Prize in Physiology and Medicine. In biophysically based neural odelling, the electrical properties of a neuron are represented in ters of an electrical equivalent circuit. Capacitors are used to odel the charge storage capacity of the cell ebrane, resistors are used to odel the various types of ion channels ebedded in ebrane, and batteries are used to represent the electrocheical potentials established by differing intracellular and extracellular ion concentrations. Figure 1 shows the equivalent circuit used by Hodgkin and Huxley in odelling a segent of squid giant axon. The current across the ebrane has two ajor coponents, one associated with the ebrane capacitance and one associated with the flow of ions through resistive ebrane channels. V in inside external stiulus I I C I Na I K I L ext C G Na G K G L ebrane V E Na E K E L V out = 0 outside Fig.1. Electrical equivalent circuit for a short segent of squid giant axon. Capacitor (capacitance C of the cell ebrane); Variable resistors (voltagedependent Na + and K + conductances G Na, G K ); fixed resistor (voltageindependent leakage conductance G L ); Batteries (reversal potentials Na +, K +, leakage - E Na, E K, E L ); Mebrane potential V = V in V out ; External stiulus I ext ; Current direction (arrows: I ext outside inside, I Na, I K and I L inside outside. Conductance G, resistance R G = 1 / R. Doing Physics With Matlab bp_hh_01 4

5 MATHEMATICAL ANALYSIS A atheatical analysis of the equivalent RC circuit for the neuron as shown in figure 1 is outlined by the following equations. (1) V = V in V out ebrane potential difference easured w.r.t. V out = 0 (2) IC dq capacitive current (displaceent current): rate of change of charge Q dt at the ebrane surface (3) Q CV Q C charge stored on surface of ebrane (4) () dv() t IC t C differentiating Q w.r.t. t at a fixed position x 0 dt (5) I INa IK IL ebrane current due to oveent of ions (6) Iext IC INa IK IL IC I Kirchhoff s current law (conservation of charge) Fundaental differential equation (D.E.) relating the change in ebrane potential to the currents across the ebrane for a sall segent of the ebrane: dv (7) C I Iext dt at a fixed position x 0 inside I 1 I2 longitudinal currents capacitive current I C I ebrane current outside I ext dx I c I I in V(t,x) V(t,x+dx) I in (t,x) I in (t,x+dx) Fig.2. Longitudinal currents inside cell, ebrane currents and capacitive current for a segent of an axon. Doing Physics With Matlab bp_hh_01 5

6 We also need to consider the current along the axon as shown in figure 2 to ake equation (7) ore general. (8) I I I I I I I I I I Kirchhoff s current law 1 2 c ext c 1 2 ext dq dv C I1 I2 I Iext currents change the charge on the ebrane dt dt (9) If we consider a sall segent (see figure 2), then I1 I2 I( x) I( x dx) I R V rin R dx V( x) V( x dx) V( x dx) V( x) I( x) I( x dx) r dx r dx in I( x) I( x dx) V( x dx) 2 V( x) V( x dx) r dx in The ebrane current crosses an area A 2 adx, using J I / A, c C / Aand using a finite difference approxiation for the second derivative of a function, we can derive a differential equation relating the ebrane voltage to the currents: in 1 ( ) 2 ( ) ( ) 1 J( x) J( x dx) 2rin dx 2r x 2 V x dx V x V x dx V 2 2 in (10) 2 V( t, x) 1 V( t, x) (, ) ext(, ) 2 t 2rin x c J t x J t x Equation (10) is a very general equation starting fro Kirchhoff s laws for a segent of the axon. The only assuptions are that the currents depend only on tie t and position x along the axon and changes on the outside of the axon can be neglected (V out = 0 for all t and x). Doing Physics With Matlab bp_hh_01 6

7 2 V ( t, x) 1 V ( t, x) (, ) ext(, ) 2 t 2 rin x c J t x J t x rate at which capacitor is gaining/losing charge per unit area ionic current densities through ebrane J = J Na + J K +J L external stiulus rate of build up of charge along inside ebrane surface because of longitudinal currents The ebrane current density depends upon driving voltages and the conductances of the ebrane. But this becoes very coplicated because the conductances depend upon the ebrane voltage V. Solutions to equation (10) are only possible by atching solutions with experiental results to deterine values of paraeters and the variation in conductances with ebrane voltage. We will consider a few special exaples in giving solutions to equation (10) that give predictions in good agreeent with experiental results. REVERSAL POTENTIALS inside [K + ] in = 140 [Na + ] in = 15 Concentration difference potential difference: reversal potential or equilibriu potential ion pup ebrane Equilibriu: Nernst equation or Boltzann factor outside [K + ] out = 5 [Na + ] out = 145 E ion k T [ Cin] loge z e [ Cout ] ion pup reversal potential teperature in kelvin [K] The nuber of ions that ove across the ebrane are sall copared to the ions in the intracellular and extracellular fluids and so the concentrations and hence reversal potential can be taken as constants. E E E Na K L 50.0 V 77.0 V 75.6 V Doing Physics With Matlab bp_hh_01 7

8 MEMBRANE CURRENT J Voltage V, current I, resistance R and conductance G are related by the equations 1 V I G R I GV J g J g V G R A A In applying these relationships to ion channels, the equilibriu (reversal) potential for each ion type also needs to be taken into account. This is the potential at which the net ionic current flowing across the ebrane would be zero for a given ion species. The reversal potentials are represented by the batteries in figure 1. Hence, (11) J g ( V E ) J g ( V E ) J g ( V E ) Na Na Na K K K L L L These relationships are coplicated because J and V are function of t and x, but also g is a function of V which is a function of t and x. We will develop a odel to give a functional relationship for each g. However, the conductance for the leakage is assued to be constant (independent of V, t or x g g ). L L inside + J K > 0 dv /dt <0 ebrane + J NA > 0 dv /dt < 0 J L > 0 dv /dt < 0 V I ext > 0 positive charges transferred fro outside to inside, depolarizing ebrane: V increases outside I ion = I > 0 positive charges transferred fro inside to outside, hyperpolarizing ebrane: V decreases Fig. 3. Sign convention for currents. A positive external current I ext (outside to inside) will tend to depolarize the cell (i.e., ake V ore positive) while a positive ionic current I ion will tend to hyperpolarize the cell (i.e., ake V = V ore negative). In a siple odel, the Na + and K + ions are considered to flow through ion channels where a series of gates deterine the conductance of the ion channel. The acroscopic conductances of the Hodgkin & Huxley odel arise fro the cobined effects of a large nuber of icroscopic ion channels ebedded in the ebrane. Each individual ion channel can be thought of as containing one or ore physical gates that regulate the flow of ions through the channel. The variation in g values is deterined by the set of gate variables n, and h. Doing Physics With Matlab bp_hh_01 8

9 K + gates: position 0n 1 Na + gates: position h 1 n n n n ion channel ebrane ion channel h gates ove to decrease conductance g K n 0 gates ove to increase conductance g K n 1 gate variables: n,, h The value of the conductance depends upon the ebrane voltage V because the values of n, and h depend on tie, their previous value at an earlier tie and the ebrane voltage. The rates of change of the gate variables are dn n n n n d dh h h h h dt dt dt (12) where the the s and s are rate constants g K g n K 4 g Na decreases: 0 g Na increases: 1 g Na decreases: h 0 g Na increases: h 1 3 gna gna h rate of closed gates opening rate of open gates closing x 1 x fraction of gates opening per second x n or or h x x fraction of gates closing per second If the ebrane voltage is claped at soe fixed value V, then values of the gate variables n, and h will reach a steady state value n, and h dx x1 x x x 0 dt where x n or or h and the steady state (equilibriu) value is given by (13) x x x x If at tie t = 0, x(0) = x 0, then a solution of equation (12) is Doing Physics With Matlab bp_hh_01 9

10 t (14) / t/ x x e n 0 e 1 x x where the tie x constant for the evolution of x is 1 (15) x x The K + channel is controlled by 4n activation gates (16) g g n n n K K x 4 dn 1 dt n The Na + channel is controlled by 3 activation gates and 1h inactivation gate 3 d 1 dh 1 dt dt (17) g g h h h K K An activation gate conductance increases with depolarization (V increases) An inactivation gate conductance decreases with depolarization (V increases) Expressions for the rate constants and were forulated by Hodgkin and Huxley in their series of voltage clap experients of giant axon nerve cells. (18) 3 T dv V V r dv (19a) n exp(1-0.1 dv) dv (19b) exp( dv) - 1 B exp(- dv / 80) n B 4 exp(- dv /18) (19c) h 0.07 exp(- dv / 20) B h 1 exp( dv) 1 The rate constants and can be calculated as functions of ebrane voltage V and teperature T using the Matlab functions alpha. and beta.. The variables V (in illivolts, V) and the teperature T (in o C) are passed onto the functions. The functions return the rate constants and (in /illiseconds, s -1 ). The resting potential V r ust be set to a global variable. The default value is V r = -65 V. Doing Physics With Matlab bp_hh_01 10

11 Matlab code for the two functions function [ An A Ah ] = alpha(v,t) % Returns rate constant in units per s (illisecond) % Inputs: V in [V] and teperature in [deg C] global Vr dv = (V - Vr); phi = 3^((T-6.3)/10); An = phi * (eps * dv)./ (eps + exp(1-0.1.* dv) - 1); A = phi * (eps * dv)./ (eps + exp( * dv) - 1); Ah = phi * 0.07.* exp(-dv./ 20); end function [ Bn B Bh ] = beta(v,t) % Returns rate constant in units per s (illisecond) % Inputs: V in [V] and teperature in [deg C] global Vr dv = (V - Vr); phi = 3^((T-6.3)/10); Bn = phi * * exp(-dv./ 80); B = phi * 4.* exp(-dv/18); Bh = phi * 1./ (exp( * dv) + 1); end The -script alpha_beta_plot. can be used to plot the rate constants and as functions of ebrane potential V at a fixed teperature T as sown in figure 4. Fig.4. Plots of the rate constants and as functions of V at T = 6.3 o C. alpha_beta_plot. Doing Physics With Matlab bp_hh_01 11

12 The steady values of (n,, h ), the steady values of (n 0, 0, h 0 ) when V = V r and the tie constants ( n,, h ) are calculated using the Matlab functions N_inf., N_0. and tau.. The result of the calculations are shown in figure 5 which was created using the Matlab - script tau_nh_inf_plot. Fig. 5. n,, h, and n,, h as functions of ebrane potential V. << n or h very quickly. n and are activating gate variables (increase in values as V increases). h is an inactivating gate variable (h decrease as V increases). The blue circle gives the value of n 0, 0 and h 0. T = 6.3 o C tau_nh_inf_plot. Doing Physics With Matlab bp_hh_01 12

13 VOLTAGE-CLAMP SIMULATIONS In any of the experients perfored by Hodgkin and Huxley, they held the ebrane at a fixed voltage by inserting an electrode into the axon of a squid. Fig. 6. Voltage-clap of a squid axon. The Matlab -script bp_neuron_02. can be used to calculate and display the voltageclap, the current densities (J, J L, J K and J Na ), the gate variables (, 3, h and n, n 4 ) and the conductances (g Na, g K ). Saple graphical outputs are shown in figure 7 for voltage claps of +20 V and +80 V. Outline the -script bp_neuron_02. structure Default resting potential V r = -65 V Voltage clap is given as a long pulse Rate constants and are calculated using the functions alpha. and beta. As the tie variable is increented, the gates variables (n,, h) then the conductances (g Na and g K ) then the current densities (J NA, J Na and J M ) are calculated for each tie step. The gate variables are calculated fro equation (12) by using the finite difference ethod to approxiate the first derivative: nt(c+1) = nt(c) + dt * (An(c) *(1-nt(c)) - Bn(c) * nt(c)); t(c+1) = t(c) + dt * (A(c) *(1-t(c)) - B(c) * t(c)); ht(c+1) = ht(c) + dt * (Ah(c) *(1-ht(c)) - Bh(c) * ht(c)); Doing Physics With Matlab bp_hh_01 13

14 Fig. 7. Variation in the gate variables, conductances and current densities for a voltage-clap applied to the axon. The depolarization produced by the clap causes a transient increase in Na + into the cell. The rise in the K + current fro the cell occurs ore slowly and is aintained as long as the ebrane is depolarized. The rate of rise of the Na+ and K+ currents increases with increasing size of the voltage clap and the peak values of Na + and K + currents are significantly increased as the clap voltage is increased, the peak values are over 100 ties the agnitudes in the resting ebrane. bp_neuron_02. Doing Physics With Matlab bp_hh_01 14

15 ACTION POTENTIAL SIMULATION The equations listed below constitute the coplete set of equations to describe the ebrane current in a squid axon and ebrane voltage. (10) 2 V( t, x) 1 V( t, x) (, ) ext(, ) 2 t 2rin x c J t x J t x (11) JNa gna ( V ENa) JK gk ( V EK) JL gl ( V EL) J JNa JK JL 4 dn 1 (16) g g n n n K K dt n 3 d 1 dh 1 dt dt (17) g g h h h K K g L g L (13) n n n n h h h h (18) 3 T dv V V r dv (19a) n exp(1-0.1 dv) dv (19b) exp( dv) - 1 B exp(- dv / 80) n B 4 exp(- dv /18) (19c) h 0.07 exp(- dv / 20) B h 1 exp( dv) 1 We will consider an axially claped axon where the interior potential does not depend upon the location x along its length: V(t,x) = V(t). Since the ebrane voltage V does not depend upon x the second derivative in equation (10) is zero (20) 2 2 V( t, x) V( t) x x Hence equation (10) siplifies to (21) Vt () c J( t) Jext ( t) t Doing Physics With Matlab bp_hh_01 15

16 If there is no external stiulus J ext = 0 and V = V r (resting potential) then J = 0 and V does not change with tie t as dv/dt = 0. It is necessary to introduce a stiulus to create a pulse. Equation (21) can be solved nuerically by using the finite difference ethod to approxiate the derivatives in equations 21, 16 and 17. An outline of the Matlab code to solve equation (21) is shown below. for cc = 1 : nu-1 [ An A Ah ] = alpha(v(cc)*1000, T); [ Bn B Bh ] = beta(v(cc)*1000, T); An = sf * An; A = sf * A; Ah = sf * Ah; Bn = sf * Bn; B = sf * B; Bh = sf * Bh; n(cc+1) = n(cc) + dt * (An *(1-n(cc)) - Bn * n(cc)); (cc+1) = (cc) + dt * (A *(1-(cc)) - B * (cc)); h(cc+1) = h(cc) + dt * (Ah *(1-h(cc)) - Bh * h(cc)); gk(cc+1) = n(cc+1)^4 * gkax; gna(cc+1) = (cc+1)^3 * h(cc+1) * gnaax; JK(cc+1) = gk(cc+1) * (V(cc) - VK); JNa(cc+1) = gna(cc+1) * (V(cc) - VNa); JL(cc+1) = gl(cc+1) * (V(cc) - VR e-3); J(cc+1) = JNa(cc+1) + JK(cc+1) + JL(cc+1); V(cc+1) = V(cc) + (dt/c)*(-jk(cc+1) - JNa(cc+1) - JL(cc+1) + Jext(cc+1)); end Doing Physics With Matlab bp_hh_01 16

17 Single current pulse The results for current pulse stiuli using the -script bp_neuron_01. are shown in the following figures. Fig. 8. The current densities for the stiulated axon at 18.5 o C. Only a very sall current pulse is required to draatically change the conductances of the ebrane to produce large K + and Na + currents. The potassiu current is positive as the K + ions ove fro inside to the outside of the cell whereas the sodiu current is negative as Na + ions ove into the cell across the ebrane. The Na + and K + currents are nearly balanced throughout ost of the pulse which lasts about 2 s. The J Na curve has an extra wiggle around t = 1.3 s caused by the rapidly changing voltage while the conductance g Na varies soothly. bp_neuron_01. Fig. 9. The conductance for potassiu and sodiu for the stiulated axon at 18.5 o C. Both the conductances vary soothly. The rise in the sodiu conductance occurs ore rapidly than for the potassiu. bp_neuron_01. Doing Physics With Matlab bp_hh_01 17

18 stiulus refractory period resting state V r = -65 V Fig. 10. Action potential produced by an external current pulse (J ext = 1.0x10-4 A.c -2 and duration 0.10 s) at a teperature of 18.5 o C. If the pulse height is halved and the pulse width is doubled ( J ext = 0.50x10-4 A.c -2 and duration 0.20 s: aount of charge transferred is constant q = I t = constant), the variation of ebrane potential with tie is alost unchanged. bp_neuron_01. J ext = 2 A.c -2 J ext = 1 A.c -2 J ext = 0.6 A.c -2 Fig. 11. Mebrane voltage responses to three different external stiuli at 18.5 o C and duration 0.10 s. (a) J ext = 0.6 A.c -2, no action potential pulse is produced, only a sall rise in the ebrane voltage and then a slow decay back to the resting potential. There is a threshold, when the the external stiulus exceeds soe critical value an action potential is produced. (b) J ext = 1.0 A.c -2, an action potential pulse is produced. (c) J ext = 2.0 A.c -2, an action potential pulse is which rises ore rapidly and to a higher peak value than the 1.0 A.c -2 stiulus. bp_neuron_01. Doing Physics With Matlab bp_hh_01 18

19 Fig. 12. Plots for the coputation of an action potential generated by a A.c -2 external stiulus in a voltage clap squid axon at 6.3 o C. The tie scale is different for the stiulus at 18.5 o C where the pulse is uch of a shorter duration. bp_neuron_01. Fig. 13. Plots for the coputation for a negative current pulse at 18.5 o C. The conductances of the ebrane decreases and the ebrane potential becoes ore polarized before slowly returning to its resting value. bp_neuron_01. Doing Physics With Matlab bp_hh_01 19

20 Multiple current pulses external stiuli 2 nd stiulus at 4.0 s action potential sae as first action potential 1 st stiulus at 0.5 s 2 nd stiulus at 3.1 s no action potential 2 nd stiulus at 3.2 s action potential but reduced peak value Fig. 14. Double stiulus. A second action potential is only produced when sufficient tie has passed for the ebrane voltage to return to nearly the resting potential. The refractory tie is about 2.6 s. If pulses occur in a tie less than 2.6 s, no action potential is generated. Doing Physics With Matlab bp_hh_01 20

21 Fig. 15. A series of current pulses can be injected into the neuron. If the repetition rate is too high for the pulses, then, not every pulse will result is an action potential being created. bp_neuron1a. Doing Physics With Matlab bp_hh_01 21

22 Step input current Doing Physics With Matlab bp_hh_01 22

23 Fig. 16. The external stiuli are step inputs for the currents densites (constant current injection). The stiuli are switched on at tie t = 5.0 s. If the size of the step is less than A.c -2 then an action potential is not produced. As the size of the step is increased, the frequency of the repetitive firing increases but the the degree of depolarization decreases. bp_neutron_01b.. Fig.17. The frequency f of the repetitive firing was deterined for each value of I 0. This was done by using the Matlab coand ginput to easure the period of the repetitive firing of the neuron in the figure window for the variation in ebrane voltage as a function of tie. bp_neuron_01bb. Doing Physics With Matlab bp_hh_01 23

24 External Stiuli with noise Noise was added to the external stiulus using the rando nubers to generate white noise. The Matlab code for the one type of noise that was used to produce the plots shown in figure 18 is: nu1 = 81; nu = 80000; Jext_ax = 0.2e-4; % ax current density for ext stiulus (A.c^-2) Jext(nu1:nu) = Jext_ax; % external stiulus current rng('shuffle'); Jext = Jext./2 + (Jext_ax./2).* (2.*rand(nu,1)-1); Fig. 18. A noisy external stiulus used to excited the neuron showing a portion of the external signal for a short tie interval. bp_neuron_01c. Doing Physics With Matlab bp_hh_01 24

25 Fig. 19. External stiuli for a rando input current density variation between 0 and 0.02 A.c -2. The input siulates noise. Spike trains are produced spasodically. The spike trains have a slightly larger tie interval between adjacent spikes than for a constant current density signal of 0.02 A.c -2 as shown in the lower plot. Doing Physics With Matlab bp_hh_01 25

26 Interval Spike Distribution Fig. 20. A step input for the current density (I 0 = 0.02 A.c -2 ) produces a spike train with action potentials produced at regular intervals (period T = 3.93 s and frequency f = 254 Hz. bp_neuron_01c. Doing Physics With Matlab bp_hh_01 26

27 Fig. 21. The ebrane voltage variation due to the noisy external stiulus shown in figure 19. bp_neuron_01c. Fig. 22. The interspike interval distribution for the spike train shown in figure 20. The spike frequency is about 191 Hz which is lower than that for the constant input shown in figure 18. bp_neuron_01c. Doing Physics With Matlab bp_hh_01 27

28 Strong stiulus and noise A neuron receives signals fro thousands of other neurons creating a noisy input resulting in sall rando fluctuations of the ebrane potential around its resting value. A strong external stiulus pulse added to the noise creates a depolarization of the ebrane producing a spike or a short spike train. Fig. 23. A sall aplitude noisy external stiulus and a strong short pulse added to the noise. The short pulse results in a firing of the neuron to produce a short spike train. bp_neuron_01c. Doing Physics With Matlab bp_hh_01 28

29 Sinusoidal external stiulus The excitation of nerve cells by sinusoidal alternating current wavefors is very dependent upon the frequency of the stiulus because of the necessity to transfer a specific aount of charge to produce the excitation. Fig. 24. A sinusoidal external current stiulus (period 5.0 s and frequency 200 Hz) produces a spike train with a frequency that atches the external stiulus. There is enough tie for the ebrane of the nerve cell to depolarize as sufficient electric charge can be applied to the ebrane within the positive half cycle of the stiulus (charge transferred equals area under current vs tie curve t2 Q i dt ). bp_neuron_01c. t1 Doing Physics With Matlab bp_hh_01 29

30 Fig. 25. A sinusoidal external current stiulus (period 2.5 s and frequency 400 Hz) does not produce a spike train. The ebrane potential oscillates with sall aplitude around the resting ebrane potential (V rest = - 65 V) with a frequency that is close to the frequency of the external stiulus. With this higher frequency of external stiulus there is not sufficient electric charge to depolarize the ebrane before the current polarity reverses which then acts to repolarize the ebrane. Fro a circuit analysis point of view, there is not sufficient tie for the capacitor to charge and hence only a sall voltage drop across it can develop. At higher frequency, the ipedance of the capacitor is low thus the voltage across it is also low. bp_neuron_01c. ap/p/iages/circuits_01.pptx Doing Physics With Matlab bp_hh_01 30