LecturesPart20 - Computational Biology, Part 20 Neuronal...

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Unformatted text preview: Computational Biology, Part 20 Neuronal Modeling Robert F. Murphy Copyright © 1996, 1999-2006. Copyright All rights reserved. Basic Neurophysiology s An imbalance of charge across a membrane An is called a membrane potential membrane s The major contribution to membrane The potential in animal cells comes from imbalances in small ions (e.g., Na, K) imbalances s The maintenance of this imbalance is an The active process carried out by ion pumps active Basic Neurophysiology s The cytoplasm of most cells (including The neurons) has an excess of negative ions over positive ions (due to active pumping of sodium ions out of the cell) sodium s By convention this is referred to as a By negative membrane potential (inside minus outside) minus s Typical resting potential is -50 mV Typical is Basic Neurophysiology s Ion pumps require energy (ATP) to carry Ion ions across a membrane up a concentration up gradient (they generate a potential) potential) s Ion channels allow ions to flow across a Ion membrane down a concentration gradient down (they dissipate a potential) dissipate Basic Neurophysiology s A cell is said to be electrically polarized cell polarized when it has a non-zero membrane potential when s A dissipation (partial or total) of the dissipation membrane potential is referred to as a depolarization, while restoration of the depolarization while resting potential is termed repolarization repolarization Basic Neurophysiology s Ion channels can switch between open and Ion open closed states closed s If an ion channel can switch its state due to If changes in membrane potential, it is said to be voltage-sensitive voltage-sensitive s A membrane containing voltage-sensitive membrane ion channels and/or ion pumps is said to be an excitable membrane excitable Basic Neurophysiology s An idealized neuron consists of An neuron x soma or cell body cell 3 contains nucleus and performs metabolic functions x dendrites 3 receive signals from other neurons through synapses receive synapses x axon 3 propagates signal away from soma x terminal branches 3 form synapses with other neurons form synapses Basic Neurophysiology vv.carleton.ca/~neil/ neural/neuron-a.html Basic Neurophysiology s Synapses pass signals from one neuron to Synapses another another s When synaptic vesicles fuse with axon When synaptic membrane, neurotransmitters are released neurotransmitters into space between axon and dendrite into s Binding of neurotransmitters to dendrite Binding causes influx of sodium ions that diffuse into soma Basic Neurophysiology s The junction between the soma and the axon The is called the axon hillock axon hillock s The soma sums (“integrates”) currents The (“inputs”) from the dendrites (“inputs”) s When the received currents result in a When sufficient change in the membrane potential, a rapid depolarization is initiated in the axon hillock hillock Basic Neurophysiology s The depolarization is caused by opening of The voltage-sensitive sodium channels that allow sodium ions to flow into the cell allow s The sodium channels only open in response The to a partial depolarization, such that a threshold voltage is exceeded is Basic Neurophysiology s As sodium floods in, the membrane As potential reverses, such that the interior is now positive relative to the outside now s This positive potential causes voltagesensitive potassium channels to open, sensitive allowing K+ ions to flow out allowing s The potential overshoots (becomes more The negative than) the resting potential negative Basic Neurophysiology s The fall in potential triggers the sodium The channels to close, setting the stage for restoration of the resting potential by sodium pumps sodium s This sequential depolarization, polarity This reversal, potential overshoot and repolarization is called an action potential action Action Potential 150 100 (uA) 50 Stimulus 0 60 40 20 0 ­20 ­40 Voltage (mV) ­60 ­80 40 G(Na) 30 G(K) 20 10 (mS/cm2) Conductance 0 0 2 4 Time (ms) 6 8 10 Basic Neurophysiology s The depolarization in the axon hillock The causes a depolarization in the region of the axon immediately adjacent to the hillock axon s Depolarization (and repolarization) Depolarization proceeds down the axon until it reaches the terminal branches terminal s The depolarization causes synaptic vesicles The to fuse with the membrane, releasing neurotransmitters to stimulate neurons with which they form synapses which Basic Neurophysiology s These sequential depolarizations form a These traveling wave passing down the axon traveling s Note that while a signal is passed down the Note axon, it is not comparable to an electrical not signal traveling down a cable signal Basic Neurophysiology s Current flows in an electrical cable x are in the direction that the signal is propagating x consist of electrons s Current flows in a neuron x are transverse to the signal propagation x consist of positively-charged ions The Hodgkin-Huxley Model s Based on electrophysiological Based measurements of giant squid axon measurements s Empirical model that predicts experimental Empirical data with very high degree of accuracy data s Provides insight into mechanism of action Provides potential potential The Hodgkin-Huxley Model s Define x v(t) ≡ v(t) voltage across the membrane at time t x q(t) ≡ net charge inside the neuron at t q(t) x I(t) ≡ current of positive ions into neuron at t I(t) x g(v) ≡ conductance of membrane at voltage v g(v) x C ≡ capacitance of the membrane capacitance x Subscripts Na, K and L used to denote specific Subscripts currents or conductances (L=“other”) currents The Hodgkin-Huxley Model s Start with equation for capacitor The Hodgkin-Huxley Model s Consider each ion separately and sum Consider currents to get rate of change in charge and hence voltage hence The Hodgkin-Huxley Model s Central concept of model: Define three state Central variables that represent (or “control”) the opening and closing of ion channels opening x m controls Na channel opening x h controls Na channel closing x n controls K channel opening The Hodgkin-Huxley Model s Define relationship of state variables to Define conductances of Na and K conductances The Hodgkin-Huxley Model s Define empirical differential equations to Define model behavior of each gate model dn = a n (v )(1 - n) - b n (v )n dt 0.01(v + 10) a n (v ) = ( v +10 ) /10 (e - 1) b n (v ) = 0.125e v / 80 The Hodgkin-Huxley Model s Define empirical differential equations to Define model behavior of each gate model dm = a m (v )(1 - m) - b m (v )m dt 0.1(v + 25) a m (v ) = ( v + 25 ) /10 (e - 1) b m (v ) = 4 e v /18 The Hodgkin-Huxley Model s Define empirical differential equations to Define model behavior of each gate model dh = a h (v )(1 - h ) - b h (v ) h dt v / 20 a h (v ) = 0.07e b h (v ) = 1 (e ( v + 30 ) /10 + 1) The Hodgkin-Huxley Model s Gives set of four coupled, non-linear, Gives ordinary differential equations ordinary s Must be integrated numerically s Constants (g in mmho/cm2 and v in mV) in g Na = 120 g K = 36 gL = 0.3 v Na = - 115 v K = 12 v L = - 10.5989 Hodgkin-Huxley Gates 150 100 (uA) 50 Stimulus 0 60 40 20 0 ­20 ­40 Voltage (mV) ­60 ­80 1.0 m gate (Na) 0.8 h gate (Na) 0.6 n gate (K) 0.4 value 0.2 Gate param 0.0 0 2 4 Time (ms) 6 8 10 Interactive demonstration s (Integration of Hodgkin-Huxley equations (Integration using Maple) using Interactive demonstration > Ena:=55: Ek:=-82: El:= -59: gkbar:=24.34: gnabar:=70.7: Ena:=55: > gl:=0.3: vrest:=-69: cm:=0.001: gl:=0.3: > alphan:=v-> 0.01*(10-(v-vrest))/(exp(0.1*(10-(v-vrest)))1): > betan:=v-> 0.125*exp(-(v-vrest)/80): > alpham:=v-> 0.1*(25-(v-vrest))/(exp(0.1*(25-(v-vrest)))-1): > betam:=v-> 4*exp(-(v-vrest)/18): > alphah:=v->0.07*exp(-0.05*(v-vrest)): > betah:=v->1/(exp(0.1*(30-(v-vrest)))+1): > pulse:=t->-20*(Heaviside(t-.001)-Heaviside(t-.002)): > rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) + > gkbar*n^4*(V-Ek) + gl*(V-El) gkbar*n^4*(V-Ek) + pulse(t))/cm: > rhsn:=(t,V,n,m,h)-> 1000*(alphan(V)*(1-n) - betan(V)*n): > rhsm:=(t,V,n,m,h)-> 1000*(alpham(V)*(1-m) - betam(V)*m): > rhsh:=(t,V,n,m,h)-> 1000*(alphah(V)*(1-h) - betah(V)*h): Interactive demonstration > inits:=V(0)=vrest,n(0)=0.315,m(0)=0.042, h(0)=0.608; > sol:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric, output=listprocedure); output=listprocedure); > Vs:=subs(sol,V(t)); > plot(Vs,0..0.02); > sol20:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric); {V(t),n(t),m(t),h(t)},type=numeric); > with(plots): Interactive demonstration > J:=odeplot(sol20,[V(t),n(t)],0..0.02): J:=odeplot(sol20,[V(t),n(t)],0..0.02): > display({J}); > pulse:=t->-2*(Heaviside(t-.001)-Heaviside(t-.002)): > rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) + gkbar*n^4*(V-Ek) + gl*(V-El)+ pulse(t))/cm: gkbar*n^4*(V-Ek) > sol2:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric); {V(t),n(t),m(t),h(t)},type=numeric); > K:=odeplot(sol2,[V(t),n(t)],0..0.02,color=green): > display({J,K}); Interactive demonstration > L:=odeplot(sol20,[V(t),n(t)],0..0.02,numpoints=400, color=blue): color=blue): > display({J,L}); > odeplot(sol20,[V(t),m(t)],0..0.02,numpoints=400); > odeplot(sol20,[V(t),h(t)],0..0.02,numpoints=400); > odeplot(sol20,[m(t),h(t)],0..0.02,numpoints=400); > a:=0.7; b:=0.8; c:=0.08; > rhsx:=(t,x,y)->x-x^3/3-y; > rhsy:=(t,x,y)->c*(x+a-b*y); > sol2:=dsolve({diff(x(t),t)=rhsx(t,x(t),y(t)), diff(y(t),t)=rhsy(t,x(t),y(t)),x(0)=0,y(0)=-1}, diff(y(t),t)=rhsy(t,x(t),y(t)),x(0)=0,y(0)=-1}, {x(t),y(t)},type=numeric, output=listprocedure); {x(t),y(t)},type=numeric, > xs:=subs(sol2,x(t)); ys:=subs(sol2,y(t)); > K:=plot([xs,ys,0..200],x=-3..3,y=-2..2,color=blue): > J:=plot({[V,(V+a)/b,V=-2.5..1.5],[V,V-V^3/3,V=-2.5..2.2]}): > plots[display]({J,K}); Interactive demonstration s (Fitzhugh-Nagamo simplification) ...
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This note was uploaded on 01/13/2012 for the course BIO 101 taught by Professor Staff during the Fall '10 term at DePaul.

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