Basic Models in Theoretical
Neuroscience
Oren Shriki
2010
Integrate and Fire and
Conductance Based Neurons
1
2
References about neurons as
electrical circuits:
• Koch, C. Biophysics of Computation, Oxford
Univ. Press, 1998.
• Tuckwell, HC. Introduction to Theoretical
Neurobiology, I&II, Cambridge UP, 1988.
3
The Neuron as an Electric Circuit
4
Intracellular Recording
5
Generation of Electric Potential on
Nerve Cell Membranes
Chief factors that determine the resting membrane potential:
• The relative permeability of the membrane to different ions
• Differences in ionic concentrations
Ion pumps – Maintain the concentration gradient by actively
moving ions against the gradient using metabolic resources.
Ion channels – “Holes” that allow the passage of ions in the
direction of the concentration gradient. Some channels are
selective for specific ions and some are not selective.
6
Ion Channels and Ion Pumps
7
The Neuron as an Electric Circuit
• Differences in ionic concentrations 
Battery
• Cell membrane  Capacitor
• Ionic channels  Resistors
8
The Neuron as an Electric Circuit
Extracellular
Intracellular
9
RC circuits
Current
source
I
C
• R – Resistance (in Ohms)
• C – Capacitance (in Farads)
R
10
RC circuits
I
C
R
• The dynamical equation is:
C
dV
dt

V
R
 I (t )
11
RC circuits
• Defining:
• We obtain:
  RC

dV
 V  I ( t )  R
dt
• The general solution is:
V t   V 0 e

t
t


R

 d t e

t t

I t  
0
12
RC circuit
• Response to a step current:
t0
I
I t   
0
t
 dt e

 t  t 

t0
I t  e
0
t

t
t
I  dt e  
0

e

t

t
I ( e  |0  e
t
V t   V 0  e


t

t

 t


I  e  1    I


t
 

 IR  1  e  
V 

t
 


1

e




13
RC circuit
• Response to a step current:
14
The Integrate-and-Fire Neuron
inside
EL
I
C
R

Threshold
mechanism
outside
• R – Membrane Resistance (1/conductance)
• C – Membrane Capacitance (in Farads)
15
Integrate-and-Fire Neuron
• If we define: V  V in  V out
• The dynamical equation will be:
dV
1
C
  V  E L   I ( t )
dt
R
• To simplify, we define: V  V in  V out  E L
• Thus:
dV
V
C
   I (t )
16
dt
R
Integrate-and-Fire Neuron
• The threshold mechanism:
– For V<θ the cell obeys its passive dynamics
– For V=θ the cell fires a spike and the voltage resets to
0.
• After voltage reset there is a refractory period, τR.
17
Integrate-and-Fire Neuron
• Response to a step current:
IR<θ:
V
t
18
Integrate-and-Fire Neuron
• Response to a step current:
IR>θ:
V
τR
τR
τR
t
T
19
Integrate-and-Fire Neuron
• Finding the firing rate as a function of the applied
current:

V  t     IR 1  e
1 e
  T  R



  T  R
e
  T  R
IR

T
R 

 

 ln  1 

IR


 

T   R   ln  1 

IR 





 1
IR
T R
 

  ln  1 

IR


20
Integrate-and-Fire Neuron
1
f I  
1

T

 R   ln  1 

 

IR 

1
 

 R   ln  1 

IR


1

R 
C
gL
 gL 

ln  1 

I


f
1
R
Ic 

R
I
21
The Hodgkin-Huxley Equations
C
dV
dt
  I ion V , w   I ( t )
I ion V , m , h , n  
g Na m h (V  V Na )  g K n (V  V K )  g L (V  V L )
3
4
dm/dt  m  (V)-m  /τ m
dh/dt  h  (V)-h  /τ h
dn/dt
 n  (V)-n  /τ n
22
The Hodgkin & Huxley Framework
C
dV
dt
  I ion V , W 1 ,  , W N   I ( t )
Each gating variable obeys the following dynamics:
dW i
dt

W V   W 
i ,
 i V
i

 i - Time constant

- Represents the effect of temperature
23
The Hodgkin & Huxley Framework
The current through each channel has the form:
I j  g j
j
V , W 1 ,  , W N (V
Vj)
g
j
- Maximal conductance (when all channels are open)

j
- Fraction of open channels
(can depend on several W variables).
24
The Temperature Parameter Φ
• Allows for taking into account different
temperatures.
• Increasing the temperature accelerates the
kinetics of the underlying processes.
• However, increasing the temperature does not
necessarily increase the excitability. Both
increasing and decreasing the temperature can
cause the neuron to stop firing.
• A phenomenological model for Φ is:
 3
 Temp
 6 . 3  / 10
25
Hodgkin & Huxley Model
dm/dt  m  (V)-m  /τ m
dh/dt  h  (V)-h  /τ h
dn/dt
 n  (V)-n  /τ n
26
Ionic Conductances During an
Action Potential
27
Repetitive Firing in Hodgkin–Huxley
Model
A: Voltage time courses in response to a step of constant
depolarizing current. from bottom to top: Iapp= 5, 15, 50, 100, 200
in μamp/cm2). Scale bar is 10 msec.
B: f-I curves for temperatures of 6.3,18.5, 26◦C, as marked.
Dotted curves show frequency of the unstable periodic orbits.
28
Fast-Slow Dissection of the
Action Potential
• n and h are slow compared to m and V.
• Based on this observation, the system can be
dissected into two time-scales.
• This simplifies the analysis.
• For details see:
Borisyuk A & Rinzel J. Understanding neuronal
dynamics by geometrical dissection of minimal models.
In, Chow et al, eds: Models and Methods in
Neurophysics (Les Houches Summer School 2003),
Elsevier, 2005: 19-72.
29
Correlation between n and h
• During the action potential the variables
n and h vary together.
• Using this correlation one can construct
a reduced model.
• The first to observe this was Fitzhugh.
30
Simplified Versions of the HH Model
• Models that generate action potentials can
be constructed with fewer dynamic
variables.
• These models are more amenable for
analysis and are useful for learning the
basic principles of neuronal excitability.
• We will focus on the model developed by
Morris and Lecar.
31
The Morris-Lecar Model (1981)
• Developed for studying the barnacle muscle.
• Model equations:
C
dV
dt
dw
dt
  I ion V , w   I app ( t )
w  V   w 

 w V 
I ion V , w   g Ca m  (V )( V  V Ca )  g K w (V  V K )  g L (V  V L )
32
Morris-Lecar Model
• The model contains K and Ca currents.
• The variable w represents the fraction of open K
channels.
• The Ca conductance is assumed to behave in an
instantaneous manner.
m  (V )  0 . 5 1  tanh
V
 V1  / V 2 
w  (V )  0 . 51  tanh
V
 V 3  / V 4 
 w V   1 / cosh V  V 3  / 2V 4 
33
Morris-Lecar Model
• A set of parameters for example:
V1   1 . 2
g Ca  4
V Ca  120
V 2  18
gK  8
V K   84
V3  2
gL  2
V L   60
μF
C  20
cm
2
  0 . 04
V 4  30
34
Morris-Lecar Model
• Voltage dependence
of the various
parameters (at long
times):
35
Conductance-Based
Models of Cortical Neurons
36
Conductance-Based Models of
Cortical Neurons
• Cortical neurons behave differently than the squid
axon that Hodgkin and Huxley investigated.
• Over the years, people developed several variations of
the HH model that are more appropriate for describing
cortical neurons.
• We will now see an example of a simple model which
will later be useful in network simulations.
• The model was developed by playing with the
parameters such that its f-I curve is similar to that of
cortical neurons.
37
Frequency-Current Responses
of Cortical Neurons
Excitatory Neuron:
Ahmed et. al., Cerebral
Cortex 8, 462-476, 1998
Inhibitory Neurons:
Azouz et. al., Cerebral
Cortex 7, 534-545, 1997
38
Frequency-Current Responses
of Cortical Neurons
The experimental findings show what f-I curves
of cortical neurons are:
• Continuous – starting from zero frequency.
• Semi-Linear – above the threshold current the
curve is linear on a wide range.
How can we reconstruct this behavior in a model?
39
An HH Neuron with a Linear f-I Curve
Shriki et al., Neural Computation 15, 1809–1841 (2003)
40
Linearization of the f-I Curve
• We start with an HH neuron that has a continuous f-I
curve (type I, saddle-node bifurcation).
• The linearization is made possible by the addition of a
certain K-current called A-current.
• The curve becomes linear only when the time constant
of the A-current is slow enough (~20 msec).
• There are other mechanisms for linearizing f-I curves.
41
Model
Equations:
Shriki et al.,
Neural
Computation
15, 1809–1841
(2003)
42