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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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