Transcript Dynamical Systems

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
7
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
8
Synaptic Dynamics: An Example
• For a single presynaptic spike the solution is:
gs(t)
g s (t )  Ge
t  s
t
9
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
11
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
12
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
13
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.
17
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.
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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.
26
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) :
27
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).
28
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
29
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
30
The Effect of Increasing gL is Subtractive
31
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]
33
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

35
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
39
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
41
A Model of a Hypercolumn in the
Primary Visual Cortex
42
‫תא‬
‫‪3‬‬
‫תא‬
‫‪2‬‬
‫תא‬
‫‪1‬‬
Neurons in the Retina and in the Thalamus
are Sensitive to Circular Spots of Light
45
Neurons in the Primary Visual Cortex are
Sensitive to Oriented Stimuli
46
Neurons in the Primary Visual Cortex are
Sensitive to Oriented Stimuli
Cell
Activity
Contrast
levels
Orientation [deg]
47
Orientation Map – A Scheme
Hypercolumn
48
Orientation Map – Experimental Results
49
What is the Mechanism Behind
Orientation Tuning?
A Feed-forward model (Hubel & Wiesel)
Thalamus
Cortex
Horizontal
Connections in
V1
51
A Model of a Hypercolumn in the
Primary Visual Cortex
52
53
54
55
56
Intermediate Summary
We constructed rate models for conductance-based networks
under two main assumptions:
a) The network is large and asynchronous.
b) The f-I curve shifts when the input conductance is changed but does not change shape.
Take Home Message: The steady-state response properties of
cortical networks may be well predicted by analytically
amenable simple rate models.
Question: How to extend this approach to study non-stationary
behavior of large neuronal systems?
57
Rate Dynamics
Two dynamical
processes:
1. The dynamics of
the firing rates
given the synaptic
activity
2. The dynamics of
the synaptic activity
given the firing
rates
58
Dynamics of synaptic
Conductances
dg (t )
g (t )

 GR(t )
dt
s
We define a normalized
synaptic activity by:
1 g (t )
r (t )  
s G
dr (t )
s
 r (t )  R(t )
dt
59
Dynamics of synaptic
Conductances
dr (t )
s
 r (t )  R(t )
dt
The synapse acts as a low-pass filter.
60
A Naive Model for the Firing
Rate Dynamics
f (t )   I (t )  I C (t ) 


  I (t )  I  Vc g L (t )
0
C


61
A Naive Model for the Firing
Rate Dynamics
• It naïve model predicts that for sinusoidal
inputs the amplitude and phase of the
response will not depend on the modulation
frequency.
• Simulations show that they do depend 
the naïve model fails!
62
Single
Neuron
Responses
to
Sinusoidal
Inputs
63
Dependence of Resonance Frequency
on Mean Firing Rate
64
A Model for the Firing Rate
Dynamics
• It is instructive to replace the instantaneous
current and conductance by filtered versions.
• Numerical simulations show that a band-pass
filter (a second order linear filter) gives a good
approximation.


f (t )   I filt (t )  I  Vc g filt (t )
0
C


65
Response to a Broadband Input
66
Excitatory Network
67
Inhibitory Network
68