Transcript Modeling Conductance-Based Networks by Rate Models

Introduction to Mathematical
Methods in Neurobiology:
Dynamical Systems
Oren Shriki
2009
Modeling Conductance-Based
Networks by Rate Models
1
References:
• Shriki, Hansel, Sompolinsky, Neural
Computation 15, 1809–1841 (2003)
• Tuckwell, HC. Introduction to Theoretical
Neurobiology, I&II, Cambridge UP, 1988.
2
Conductance-Based Models vs. Simplified Models
• There are two main classes of theoretical approaches
to the behavior of neural systems:
– Simulations of detailed biophysical models.
– Analytical (and numerical) solutions of simplified models (e.g.
Hopfield models, rate models).
• Simplified models are extremely useful for studying the
collective behavior of large neuronal networks.
However, it is not always clear when they provide a
relevant description of the biological system, and what
meaning can be assigned to the quantities and
parameters used in them.
3
Conductance-Based Models vs. Simplified Models
• Using mean-field theory we can describe the dynamics
of a conductance-based network in terms of firing-rates
rather than voltages.
• The analysis will lead to a biophysical interpretation of
the parameters that appear in classical rate models.
• The analysis will be divided into two parts:
– A. Steady-state analysis (constant firing rates)
– B. Firing-rate dynamics
4
Network Architecture
External Inputs (independent
Poisson processes)
f1inp
fNinp
Recurrent
connectivity
f1
fN
1
2
3
4
N5
Voltage Dynamics for A Network of
Conductance-Based Point Neurons
We assume that the neurons are point neurons obeying
Hodgkin-Huxley type dynamics:
dVi
Cm
  g L Vi (t )  EL   I iactive  I iext  I inet  I iapp
dt
•
•
•
•
(i  1, , N )
Iactive – Ionic current involved in the action potential
Iext – External synaptic inputs
Inet – Synaptic inputs from within the network
Iapp – External current applied by the experimentalist
6
Synaptic Conductances
A spike at time tspike of the presynaptic cell contributes to
the postsynaptic cell a time-depndent conductance , gs(t):
gs t 
Peak
conductance
G
s
t_spike
t
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Synaptic Dynamics: An Example
• Synaptic dynamics are usually characterized by
fast rise and slow decay.
• The simplest model assumes instantaneous rise
and exponential decay:
dg s
gs
   GR(t )
dt

R(t )    t  t spike 
(Presynaptic rate)
t spike
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Synaptic Dynamics: An Example
• For a single presynaptic spike the solution is:
gs(t)
g s (t )  Ge
t  s
t
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Synaptic Dynamics: An Example
• Implementation in numerical simulations:
• Given the time step dt define the attenuation
factor:
edt  e
 dt  s
• A dimensionless parameter, f, is increased by 1
after each presynaptic spike and multiplied by
the attenuation factor in each time step.
• The conductance is the product of f and the peak
conductance, G.
10
Synaptic Dynamics
• For simplicity, we shall write in general:
g s (t )  GK (t )
• K(t) is the time course (dimensionless) function.
• We define:

 s   K (t )dt
0
• For example:

e
0
t  s

dt   s e

t  s 
0
s
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External Synaptic Current
• The explicit expression for the external synaptic
current is:

I iext (t )  g iinp (t ) E inp  Vi (t )

• The peak synaptic conductance is:
Giinp
• The time constant is:
 iinp
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Internal Synaptic Current
• The explicit expression for the internal synaptic
current is:
I (t )   gij (t )E j  Vi (t )
N
net
i
j 1
• The peak synaptic conductance is:
Gij
• The time constant is:
 ij
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Part A: Steady-State Analysis
The main assumptions are:
• Firing rates of external inputs are constant in time
• Firing rates within the network are constant in time
• The network state is asynchronous
• The network contains many neurons
14
Asynchronous States in Large Networks
• In large asynchronous networks each neuron is
bombarded by many synaptic inputs at any moment.
• The fluctuations in the total input synaptic conductance
around the mean are relatively small.
• Thus, the total synaptic conductance in the input to each
neuron is (approximately) constant.
15
Asynchronous States in Large Networks
• The figure below shows the synaptic conductance of a
certain postsynaptic neuron in a simulation of two
interacting populations (excitatory and inhibitory):
16
Mean-Field Approximation
• We can substitute the total synaptic conductance by its
mean value.
• This approximation is called the “Mean-Field” (MF)
Approximation.
• The justification for the MF approximation is the central
limit theorem.
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The Central Limit Theorem
• A random variable ,which is the sum of many
independent random variables, has a Gaussian
distribution with mean value equal to the sum of the
mean values of the individual random variables.
• The ratio between the standard deviation and the mean of
the sum satisfies:
std
1

mean
N
where N is the number of individual random variables.
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Mean-Field Analysis of the Synaptic Inputs
• The total contribution to the i’th neuron from within the
network is:
g inet (t )   g ij (t )   Gij K ij t  t j 
N
N
j 1
j 1 t j
• The contribution to this sum from the j’th neuron is:
gij (t )   Gij K ij t  t j 
tj
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Mean-Field Analysis of the Synaptic Inputs
• Consider a time window T>>1/fj, where fj is the firing
rate of the presynaptic neuron j.
• Schematically, the contribution of the j’th neuron in this
time window looks like this:
spikes
t
T
g ij (t )
g ij
t
T
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Mean-Field Analysis of the Synaptic Inputs
• The mean conductance due to the j’th neuron over a long
time window is:
T

N spike (T )
1
g ij   g ij (t )dt 
 Gij   K ij (t )dt
T0
T
0
 Gij ij f j
• The mean conductance resulting from all neurons in the
network is:
g
net
i
N
N
j 1
j 1
  gij  Gij ij f j
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Mean-Field Analysis of the Synaptic Inputs
• The mean-field approximation is:
N
ginet (t )  ginet   Gij ij f j
j 1
• Effectively, we replace a spatial averaging by a temporal
averaging .
• Using a similar analysis, the contribution of the external
inputs can be replaced by:
g iext (t )  g iext  Giinp iinp f i inp
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Mean-Field Analysis of the Synaptic Inputs
• The synaptic currents are not constant in time since the
voltage of the neuron varies significantly over time:
I inet (t )   gij (t )E j  Vi (t )
N
j 1
• We can decompose the last expression in the following
way:
I inet (t )   g ij (t )E j  EL  EL  Vi (t ) 
N
j 1
  g ij (t )E j  EL   Vi (t )  EL  g ij (t )
N
N
j 1
j 1
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Mean-Field Analysis of the Synaptic Inputs
• We now use the MF approximation :
N
N
j 1
j 1
net
g
(
t
)

g
 ij
i (t )   Gij ij f j
 g (t )E
N
j 1
ij
 EL    Gij ij f j E j  EL 
N
j
j 1
• This gives:
I inet (t )   Gij ij E j  EL  f j  Vi (t )  EL  Gij ij f j
N
N
j 1
j 1
A constant applied
current
A constant increase in the
leak conductance
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Mean-Field Analysis of the Synaptic Inputs
• Similarly:


I iext (t )  Giinp iinp Eiinp  EL f i inp  Vi (t )  EL Giinp iinp f i inp
• We obtained the following mapping:


I iapp  I iapp   Gij ij E j  EL  f j  Giinp iinp Eiinp  EL f i inp
N
j 1
N
g L  g L   Gij ij f j  Giinp iinp f i inp
j 1
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Mean-Field Analysis of the Synaptic Inputs
• To sum up:
• Asynchronous Synaptic Inputs Produce a
Stationary Shift in the
Voltage-Independent Current
and in the
Input Passive Conductance
of the Postsynaptic Cell.
• To complete the loop and determine the network’s
firing rates we need to know how the firing rate of
a single cell is affected by these shifts.
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Current-Frequency Response Curves of
Cortical Neurons are Semi-Linear
Excitatory Neuron (After:
Ahmed et. al., Cerebral
Cortex 8, 462-476, 1998):
Inhibitory Neurons (After:
Azouz et. al., Cerebral
Cortex 7, 534-545, 1997) :
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The Effect of Changing the Input Conductance is Subtractive
Experiment:
f-I curves of a cortical neuron before and after iontophoresis
of baclofen, which opens synaptic conductances.
(Connors B. et. al., Progress in Brain Research, Vol. 90, 1992).
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A Hodgkin-Huxley Neuron with an A-current
dV
C
  I ion V , w  I (t )
dt
I ion V , m, h, n 
g Na m3 h(V  ENa )  g K n 4 (V  EK )  g Aa3 b(V  EK )  g L (V  EL )
dh/dt  h(V)-h /τ h
dn/dt  n(V)-n /τ n
db/dt  b(V)-m /τ b
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The Addition of a Slow Hyperpolarizing Current
Produces a Linearization of the f-I Curve
gA=20, A=20, I=1.6 [A/cm2]
____ - No A-current (gA=0)
____ - Instantaneous A-current (gA=20, A=0)
____ - Slow A-current (gA=20, A=20)
[gA]=mS/cm2, [A]=msec
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The Effect of Increasing gL is Subtractive
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Model
Equations
with
Parameters:
Shriki et al.,
Neural
Computation
15, 1809–1841
(2003)
32
The Dependence of the Firing Rate on I and on gL Can
Be Described by a Simple Phenomenological Model
f   I  I C  
 
  I  I  Vc g L
0
C


[x]+=x if x>0 and 0 otherwise.
We find for the model neuron :
=35.4 [Hz/(A/cm2)]
Vc=5.6 [mV]
IC0=0.65 [A/cm2]
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The Effect of the Synaptic Input On the
Firing Rate
Combining the previous results, we find that the
steady-state firing rates obey the following
equations:
 


f i   I iapp  Giinp iinp Eiinp  EL  Vc f i inp 

Gij ij E j  EL  Vc  f j  I  Vc g L 

j 1

N

0
c

34
The Effect of the Synaptic Input On the
Firing Rate
This can be written us:
N
 app

inp inp
f i    I i  J i f i   J ij f j  I c 
j 1


Where:
 
J ij  Gij ij E j  EL  Vc 

J iinp  Giinp iinp Eiinp  EL  Vc

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The Units of the Interactions
• The units of J are units of electric charge:
J   g t V   I t   Q  C
• The quantity Jijfj reflects the mean current due
to the j’th synaptic source.
• The strength of the interaction Jij reflects the
amount of charge that is transferred with each
presynaptic action potential.
36
The Sign of the Interaction
• The interaction strength in the rate model has
the form:
J ij  GijE ijE  Es  EL  Vc 
• The rule for excitation / inhibition is:
Es  EL  Vc
‘excitatory’
Es  EL  Vc
‘inhibitory’
• This does not necessarily coincide with the
biological definition of excitation/inhibition.
37
The Sign of the Interaction
• The biological definition is:
Es  
excitatory
Es  
inhibitory
• A positive J implies that this synaptic source
increases the firing rate.
• In general, it may be that a certain synaptic source
tends to elicit a spike but increases the
conductance in a way that reduces the overall
firing rate.
38
The Model Parameters
• Neuron:
• β – Slope of frequency-current response
• Vc, Ic0 – Dependence of current threshold on leak
conductance
• EL – Reversal potential of leak conductance
• Synapse:
• Gij – Peak synaptic conductance
• Ej – Synaptic reversal potential
• τij – Synaptic time constant
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Rate Model for a Homogeneous,
Highly Connected Excitatory
Network:

f  J
inp
f
inp
 N c Jf  I c



inp inp

f 
J f  Ic 
1  N c J
40
Simulations of the Excitatory Network and
Rate Model Prediction
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