Lecture 10: Mean Field theory with fluctuations and correlations Reference: A Lerchner et al, Response Variability in Balanced Cortical Networks, q-bio.NC/0402022,

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Presentation transcript:

Lecture 10: Mean Field theory with fluctuations and correlations Reference: A Lerchner et al, Response Variability in Balanced Cortical Networks, q-bio.NC/ , Neural Computation (also on course webpage)

Mean field theory for disordered systems In network with fixed randomness (here: random connections):

Mean field theory for disordered systems In network with fixed randomness (here: random connections): =>

Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs =>

Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents =>

Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents =>

Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents  Different rates =>

Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents  Different rates => Temporal fluctuations:

Mean field theory for disordered systems In network with fixed randomness (here: random connections): Different neurons have different number of inputs  Different neurons have different mean input currents  Different rates => Have to include these fluctuations in the theory => Temporal fluctuations:

Heuristic treatment Start from(one population for now)

Heuristic treatment Start from(one population for now)

Heuristic treatment Start from Time average: (one population for now)

Heuristic treatment Start from Time average: Average over neurons: (one population for now)

Heuristic treatment Start from Time average: Average over neurons: Neuron-to-neuron fluctuations: (one population for now)

Temporal fluctuations Input current fluctuations:

Temporal fluctuations Input current fluctuations: Correlations:

Temporal fluctuations Input current fluctuations: Correlations: Have to calculate (“order parameters”) self-consistently

2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky)

2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) 0 1 2

2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) Mean number of connections from population b to a neuron in population a : 0 1 2

2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) Mean number of connections from population b to a neuron in population a : 0 1 2

2-population model Like Amit-Brunel model, but with different scaling of synapses (van Vreeswijk-Sompolinsky) Populations 0,1,2 (as before) Mean number of connections from population b to a neuron in population a : 0 1 2

Means and variances of synaptic strengths

Mean:

Means and variances of synaptic strengths Mean: Variance:

Input current statistics Mean (average of time and neurons)

Input current statistics Mean (average of time and neurons)

Input current statistics Mean (average of time and neurons) Neuron-to-neuron fluctuations of the temporal mean:

Input current statistics Mean (average of time and neurons) Neuron-to-neuron fluctuations of the temporal mean:

Input current statistics Mean (average of time and neurons) Neuron-to-neuron fluctuations of the temporal mean: Temporal fluctuations:

Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current

Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current

Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current with

Equivalent single-neuron problem Single neuron (excitatory or inhibitory) driven by input current with

Can combine the two kinds of fluctuations: Consider the total correlation function

Can combine the two kinds of fluctuations: Consider the total correlation function

Can combine the two kinds of fluctuations: Consider the total correlation function Average over neurons

Can combine the two kinds of fluctuations: Consider the total correlation function Average over neurons Total input current fluctuations:

Balance condition Total average current:

Balance condition Total average current: i.e., with

Balance condition Total average current: i.e., with

Balance condition Total average current: i.e., with or, defining

Balance condition Total average current: i.e., with or, defining

Balance condition Total average current: i.e., with or, defining Solution:

Balance condition Total average current: i.e., with or, defining Solution: i.e., can solve for mean rates independent of fluctuations/correlations

Correlations/fluctuations Have to do it numerically:

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function Simulate many trials with different realizations of noise, collect statistics (measure r a, q a, C a (t) )

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function Simulate many trials with different realizations of noise, collect statistics (measure r a, q a, C a (t) ) Use these to generate improved noise samples

Correlations/fluctuations Have to do it numerically: Start with r b from balance eqn, initial estimates Generate Gaussian noisy input current with mean and correlation function Simulate many trials with different realizations of noise, collect statistics (measure r a, q a, C a (t) ) Use these to generate improved noise samples Simulate again, repeat until input and output order parameters agree

When done, compute firing statistics of single neurons: For a single neuron, with effective input current

When done, compute firing statistics of single neurons: For a single neuron, with effective input current Have to hold x ab fixed

Experimental background: firing statistics Gershon et al, J Neurophysiol 79, (1998) x x x x x x

Experimental background: firing statistics Gershon et al, J Neurophysiol 79, (1998) Variance > mean x x x x x x

Experimental background: firing statistics Gershon et al, J Neurophysiol 79, (1998) Variance > mean x x x x x x i,e., Fano factor F > 1

Fano factors and correlation functions Spike count:

Fano factors and correlation functions Spike count: Mean:

Fano factors and correlation functions Spike count: Mean: Variance:

Fano factors and correlation functions Spike count: Mean: Variance: => Fano factor

Model calculations Synaptic matrix:

Model calculations Synaptic matrix: J s = J s = 0.75 J s = 1.5 Correlation functions: g = 1

Interspike interval distributions Js = 1.5 Js = 0.75 J s = 0.375

Spike count variance vs mean

What controls F? Membrane potential distributions have width ~ J ab = O(1)

What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset Membrane potential distributions have width ~ J ab = O(1)

What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset => F < 1 Membrane potential distributions have width ~ J ab = O(1)

What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset => F < 1 Reset near threshold: initial spread of membrane potential distribution Membrane potential distributions have width ~ J ab = O(1)

What controls F? Low post-spike reset voltage: takes time (~  ) to recover from reset => F < 1 Reset near threshold: initial spread of membrane potential distribution => excess early spikes, F > 1 Membrane potential distributions have width ~ J ab = O(1)