Lecture 11: Networks II: conductance-based synapses, visual cortical hypercolumn model References: Hertz, Lerchner, Ahmadi, q-bio.NC/ [Erice lectures] Lerchner, Ahmadi, Hertz, q-bio.NC/ (Neurocomputing, 2004) [conductance-based synapses] Lerchner, Sterner, Hertz, Ahmadi, q-bio.NC/ [orientation hypercolumn model]

Conductance-based synapses In previous model:

Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance:

Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance:

Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train

Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train kernel:

Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train kernel:

Conductance-based synapses In previous model: But a synapse is a channel with a (neurotransmitter-gated) conductance: whereis the synaptically-filtered presynaptic spike train kernel:

Model

Mean field theory Effective single-neuron problem with synaptic input current

Mean field theory Effective single-neuron problem with synaptic input current

Mean field theory Effective single-neuron problem with synaptic input current with

Mean field theory Effective single-neuron problem with synaptic input current with where = correlation function of synaptically-filtered presynaptic spike trains

Balance condition Total mean current = 0 :

Balance condition Total mean current = 0 :

Balance condition Total mean current = 0 : Mean membrane potential just below 

Balance condition define Total mean current = 0 : Mean membrane potential just below 

Balance condition define Total mean current = 0 : Mean membrane potential just below 

Balance condition define Solve for r b as in current-based case: Total mean current = 0 : Mean membrane potential just below 

Balance condition define Solve for r b as in current-based case: Total mean current = 0 : Mean membrane potential just below 

Balance condition define Solve for r b as in current-based case:  Total mean current = 0 : Mean membrane potential just below 

High-conductance-state

V a “chases” V s a (t) at rate g tot (t)

High-conductance-state V a “chases” V s a (t) at rate g tot (t)

High-conductance-state V a “chases” V s a (t) at rate g tot (t)

High-conductance-state V a “chases” V s a (t) at rate g tot (t) Effective membrane time constant ~ 1 ms

Membrane potential and spiking dynamics for large g tot

Fluctuations Measure membrane potential from :

Fluctuations Measure membrane potential from :

Fluctuations Measure membrane potential from : Conductances: mean + fluctuations:

Fluctuations Measure membrane potential from : Conductances: mean + fluctuations:

Fluctuations Measure membrane potential from : Conductances: mean + fluctuations:

Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations:

Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: =>

Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: => or

Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: => or with

Fluctuations Measure membrane potential from : Use balance equation in Conductances: mean + fluctuations: => or with

Effective current-based model High connectivity:

Effective current-based model High connectivity:

Effective current-based model High connectivity:

Effective current-based model High connectivity:

Effective current-based model High connectivity:

Effective current-based model High connectivity: Like current-based model with

Effective current-based model High connectivity: Like current-based model with (but effective membrane time constant depends on presynaptic rates)

Firing irregularity depends on reset level and  s

Modeling primary visual cortex

Background: 1.Neurons in primary visual cortex (area V1) respond strongly to oriented stimuli (bars, gratings)

Modeling primary visual cortex Background: 1.Neurons in primary visual cortex (area V1) respond strongly to oriented stimuli (bars, gratings)

Modeling primary visual cortex Background: 1.Neurons in primary visual cortex (area V1) respond strongly to oriented stimuli (bars, gratings) Note: contrast- invariant tuning width

Spatial organization of area V1 2. In V1, nearby neurons have similar orientation tuning

Spatial organization of area V1 2. In V1, nearby neurons have similar orientation tuning

Orientation column ~ 10 4 neurons that respond most strongly to a particular orientation

Orientation column ~ 10 4 neurons that respond most strongly to a particular orientation Tuning of input from LGN (Hubel-Wiesel):

Hubel-Wiesel feedforward connectivity cannot by itself explain contrast-invariant tuning Simplest model: cortical neurons sums H-W inputs, firing rate is threshold-linear function of sum

Hubel-Wiesel feedforward connectivity cannot by itself explain contrast-invariant tuning Simplest model: cortical neurons sums H-W inputs, firing rate is threshold-linear function of sum

Modeling a “hypercolumn” in V1 Coupled collection of networks, each representing an “orientation column”

Modeling a “hypercolumn” in V1 Coupled collection of networks, each representing an “orientation column”

Modeling a “hypercolumn” in V1 Coupled collection of networks, each representing an “orientation column”

Modeling a “hypercolumn” (2)

 0 is stimulus orientation

Modeling a “hypercolumn” (2)  0 is stimulus orientation (simplest model periodic in  with period  )

Modeling a “hypercolumn” (2)  0 is stimulus orientation (simplest model periodic in  with period  )

Modeling a “hypercolumn” (2)  0 is stimulus orientation (simplest model periodic in  with period  )

Modeling a “hypercolumn” (2)  0 is stimulus orientation Connection probability falls off with increasing  ’, reflecting probable greater distance. (simplest model periodic in  with period  )

Mean field theory Effective intracortical input current

Mean field theory Effective intracortical input current mean

Mean field theory Effective intracortical input current mean fluctuations:

Mean field theory Effective intracortical input current mean fluctuations: with

Mean field theory Effective intracortical input current mean fluctuations: with Solve self-consistently for order parameters

Balance condition Total mean current vanishes at all  :

Balance condition Total mean current vanishes at all  :

Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation:

Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation:

Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation: Integral equations for r a (  )

Balance condition Total mean current vanishes at all  : Ignore leak, make continuum approximation: Integral equations for r a (  ) Can take  0 = 0

Broad tuning

Make ansatz

Broad tuning Make ansatz use

Broad tuning Make ansatz use

Broad tuning Make ansatz use => with

Broad tuning Make ansatz use => with Solve for Fourier components:

Broad tuning Make ansatz use => with Solve for Fourier components:

Broad tuning Make ansatz use => with Solve for Fourier components: Valid for (otherwise r a (  ) < 0 at large  )

 narrow tuning useonly for

 narrow tuning useonly for i.e.,

 narrow tuning useonly for i.e., same  c for both populations – consequence of

 narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in

 narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in and same  for all interactions in

 narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in and same  for all interactions in Balance condition:

 narrow tuning useonly for i.e., same  c for both populations – consequence of same  for both populations in and same  for all interactions in Balance condition: =>

Narrow tuning (2) Now do the integrals:

Narrow tuning (2) Now do the integrals:

Narrow tuning (2) Now do the integrals: where

Narrow tuning (2) Now do the integrals: where f0:f2:f0:f2: ______

Narrow tuning (3)

Divide one by the other:

Narrow tuning (3) Divide one by the other: determines  c

Narrow tuning (3) Divide one by the other: determines  c  c is independent of I a0 : contrast-invariant tuning width (as in experiments)

Narrow tuning (3) Divide one by the other: determines  c  c is independent of I a0 : contrast-invariant tuning width (as in experiments) Then can solve for rate components:

Narrow tuning (3) Divide one by the other: determines  c  c is independent of I a0 : contrast-invariant tuning width (as in experiments) Then can solve for rate components:

Noise tuning Input noise correlations:

Noise tuning Input noise correlations:

Noise tuning Input noise correlations:

Noise tuning Input noise correlations: =>

Noise tuning Input noise correlations: => Same integrals as in rate computation =>

Noise tuning Input noise correlations: => Same integrals as in rate computation =>

Noise tuning Input noise correlations: => Same integrals as in rate computation => using

Noise tuning Input noise correlations: => Same integrals as in rate computation => using=>

Noise tuning Input noise correlations: => Same integrals as in rate computation => using=>Same tuning as input!

Some numerical results (1)

Numerical results (2): Fano factor tuning

Numerical results (3): noise tuning vs firing tuning