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)