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BioE332 Lecture 3: Balanced networks - Stanford University · MT neurons spike-counts Spikes counts...
Transcript of BioE332 Lecture 3: Balanced networks - Stanford University · MT neurons spike-counts Spikes counts...
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BioE332 Lecture 3: Balanced networks
Kwabena BoahenSpring 2013
Friday, April 19, 13
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Poisson process
A: Probability that 0, 1, 2, and 5 spikes occur in a period T when the mean rate is r.
B: Probability n spikes occur in a period T=10/r — approximately Gaussian for rT>>1.
Dayan & Abbott ’01
10
Exponential interval distribution Variance = mean
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Defining propertiesEvent counts’ variability is measured by:
σ n2 = n = rT
Fano factor ≡ σ n2
n
CV ≡στ
τ
τ =1rand στ
2 =1r2
For a Poisson process:
Event intervals’ variability is measured by:
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MT neurons spike-counts
Spikes counts of responses to moving images are Poissonian (alert macaque).
O’Keefe et al., ’97
A: Variance versus mean for spike counts in 256ms bins. B,C: Multiplier and exponent for bins of different sizes.
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MT neuron spike-intervals
Interspike intervals are Poissonian, except for very short intervals — fit by including refractory period (or with Gamma distribution).
Bair et al., ’94
A: Measured interspike interval distribution (non-bursty cell). B: Poisson model with a stochastic refractory period (Gaussian
with 5ms mean and 2ms SD).
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Balanced regimeExcitatory and inhibitory inputs are balanced so as to keep the cell’s average potential close to threshold.More formally, each cell is connected to K others, but sqrt(K) spikes suffice to bring it to threshold.In this regime, small fluctuations in the input drive spiking, and the firing pattern is strongly Poisson-like. Note that the network’s connectivity must be random and sparse (K<<N, the total).
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Integrate-and-fire model
Leaky integrate-and-fire neurons with noisy input can fire regularly (A) or irregularly (B). Depends on whether the mean
free membrane potential (top) is above or below threshold.
Dayan & Abbott ’01
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The question is: “Can balance between
excitation and inhibition emerge
without fine tuning?”Friday, April 19, 13
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Balanced network studiesIn networks of binary neurons, balance emerges dynamically through strong negative feedback, which also linearizes the response to external inputs (Van Vreeswijk & Sompolinsky ‘96, ‘98). In networks of leaky integrate-and-fire neurons with delta synapses, regular and irregular synchronous states exists, in addition to regular and irregular asynchronous states (Brunel ‘00). These four states also exists in networks of cortex-like locally connected random networks with alpha synapses (Mehring et al., 2003).
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Binary neuronsBinary-valued output (0 or 1) is updated by choosing neurons sequentially, in random order.Time between consecutive updates of a unit represents the membrane time constant (10ms).
Neuron model
Synapse model
σ i (t) = Θ ui (t)( )where Θ x( ) ≡ 1 if x > 00 if x ≤ 0
⎧⎨⎪
⎩⎪
ui (t) = Jijσ j (t) + u0
j∑ −θ
ui (t) σ i (t)
σ j (t) J ji
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Binary neuron networkEach neuron receives K excitatory, inhibitory, and K external connections, chosen randomly such that K<<NE,NI,N0.The threshold θ is chosen such that only √K excitatory inputs are required to cross it.The total synaptic current is approximately Gaussian with:µk (t) = K Jklml (t)
l∈E , I ,0∑ σ k
2 (t) = K (Jkl )2
l∈E , I∑ ml (t)
where ml(t) is the average activity of population l.
m0 (t)NE
NI
JEE = J
JE0 = J
JEI = −gJ
JII = −gJ
JIE = J
JI 0 = Jm0 (t)
mE (t)
N0
mI (t)
J ≈ 1,g ≈ 2,K = 1000,D = 0.3
K < θE , I < (1+ D) K
Parameters values
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Model’s balanced state
Top to bottom: Total excitatory input (external and feedback), net input,
total inhibitory input, and a neuron’s activity.
Mean activities of excitatory (solid line) and inhibitory (dashed line)
units increase with mean activity of external units.
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Firing-rate distribution
Distribution is long-tailed, more so for lower (solid
line) than higher (dashed line) population-averaged
rates (mE=0.01,0.03).
Distribution in prefrontal cortex neurons in a
monkey attending to stimuli and executing reaching movements.
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Is it fine-tuned?How much variation in threshold (relative to the total amount of input) can the model tolerate?
ε = D KK
≈1K
This is 1% precision for K=10,000, the typical number of inputs a cortical cell receives.
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Stimulus onset quenches
variability
Fano factor is greater than oneDrops to around one when stimulus comes on
Churchland et al. ’10Friday, April 19, 13
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Doubly stochastic processSchadlen et al. ’11
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LIP Neurons Making Decision Schadlen et al. ’11
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Clustered balanced network
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Uniform vs Clustered
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Rate and Fano Factor Distribution
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Reproduces drop in Fano Factor
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