Neuronal Dynamics: Computational Neuroscience of Single Neurons
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1 Week 5 part 3a :Three definitions of rate code Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 5 Variability and Noise: The question of the neural code Wulfram Gerstner EPFL, Lausanne, Switzerland 5.1 Variability of spike trains - experiments 5.2 Sources of Variability? - Is variability equal to noise? 5.3 Three definitions of Rate code - Poisson Model 5.4 Stochastic spike arrival - Membrane potential fluctuations 5.5. Stochastic spike firing - subthreshold and superthreshold
2 Week 5 part 3a :Three definitions of rate code 5.1 Variability of spike trains - experiments 5.2 Sources of Variability? - Is variability equal to noise? 5.3 Three definitions of Rate code - Poisson Model 5.4 Stochastic spike arrival - Membrane potential fluctuations 5.5. Stochastic spike firing - subthreshold and superthreshold
3 Neuronal Dynamics 5.3. Three definitions of Rate Codes 3 definitions - Temporal averaging - Averaging across repetitions - Population averaging ( spatial averaging)
4 Neuronal Dynamics 5.3. Rate codes: spike count Variability of spike timing trial 1 rate as a (normalized) spike count: Brain stim ν () t = T=1s sp n T single neuron/single trial: temporal average
5 Neuronal Dynamics 5.3. Rate codes: spike count single neuron/single trial: temporal average ν () t = Variability of interspike intervals (ISI) measure regularity 100 u [mv] sp n T ISI ISI distribution time [ms] t [ms] 500
6 Neuronal Dynamics 5.3. Spike count: FANO factor trial 1 trial 2 n = sp 1 5 sp n 2 = 6 trial K sp n K = 4 Brain stim T Fano factor F = ( sp sp n ) 2 k nk n sp k
7 Neuronal Dynamics 5.3. Three definitions of Rate Codes 3 definitions - Temporal averaging (spike count) ISI distribution (regularity of spike train) Fano factor (repeatability across repetitions) - Averaging across repetitions - Population averaging ( spatial averaging) Problem: slow!!!
8 Neuronal Dynamics 5.3. Three definitions of Rate Codes 3 definitions - Temporal averaging Problem: slow!!! - Averaging across repetitions - Population averaging
9 Neuronal Dynamics 5.3. Rate codes: PSTH Variability of spike timing trial 1 trial 2 trial K Brain stim
10 Neuronal Dynamics 5.3. Rate codes: PSTH Averaging across repetitions single neuron/many trials: average across trials PSTH ( t) = n( t; t + Δt) K Δt K repetitions t Stim(t) PSTH(t) K=50 trials
11 Neuronal Dynamics 5.3. Three definitions of Rate Codes 3 definitions - Temporal averaging - Averaging across repetitions Problem: not useful for animal!!! - Population averaging
12 Neuronal Dynamics 5.3. Rate codes: population activity population of neurons with similar properties neuron 1 neuron 2 Neuron K Brain stim
13 Neuronal Dynamics 5.3. Rate codes: population activity population activity - rate defined by population average t natural population activity A( t) = n( t; t + Δt) NΔt t
14 Neuronal Dynamics 5.3. Three definitions of Rate codes Three averaging methods - over time Too slow for animal!!! - over repetitions Not possible for animal!!! - over population (space) natural
15 Neuronal Dynamics Quiz 5.2. Rate codes. Suppose that in some brain area we have a group of 500 neurons. All neurons have identical parameters and they all receive the same input. Input is given by sensory stimulation and passes through 2 preliminary neuronal processing steps before it arrives at our group of 500 neurons. Within the group, neurons are not connected to each other. Imagine the brain as a model network containing nonlinear integrate-and-fire neurons, so that we know exactly how each neuron functions. Experimentalist A makes a measurement in a single trial on all 500 neurons using a multielectrode array, during a period of sensory stimulation. Experimentalist B picks an arbitrary single neuron and repeats the same sensory stimulation 500 times (with long pauses in between, say one per day). Experimentalist C repeats the same sensory stimulation 500 times (1 per day), but every day he picks a random neuron (amongst the 500 neurons). All three determine the time-dependent firing rate. [ ] A and B and C are expected to find the same result. [ ] A and B are expected to find the same result, but that of C is expected to be different. [ ] B and C are expected to find the same result, but that of A is expected to be different. [ ] None of the above three options is correct.
16 Week 5 part 3b :Poisson Model Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 5 Variability and Noise: The question of the neural code Wulfram Gerstner EPFL, Lausanne, Switzerland 5.1 Variability of spike trains - experiments 5.2 Sources of Variability? - Is variability equal to noise? 5.3 Three definitions of Rate code - Poisson Model 5.4 Stochastic spike arrival - Membrane potential fluctuations 5.5. Stochastic spike firing - subthreshold and superthreshold
17 Week 5 part 3b :Poisson Model 5.1 Variability of spike trains - experiments 5.2 Sources of Variability? - Is variability equal to noise? 5.3 Three definitions of Rate code - Poisson Model 5.4 Stochastic spike arrival - Membrane potential fluctuations 5.5. Stochastic spike firing - subthreshold and superthreshold
18 Neuronal Dynamics 5.3b. Inhomogeneous Poisson Process t A(t) I(t) T n sp PSTH ( t) = n( t; t K + Δt) Δt n( t; t + Δt) A( t) = NΔt population activity Pure rate code = stochastic spiking à Poisson model
19 Neuronal Dynamics 5.3b. Poisson Model Homogeneous Poisson model: constant rate Math detour: Poisson model Δt Probability of finding a spike PF = ρ Δt 0 Pure rate code = stochastic spiking à Poisson model
20 Neuronal Dynamics 5.3b. Poisson Model Probability of firing: P F = ρ Δt 0 Δt Take Δt 0
21 Neuronal Dynamics 5.3b. Interval distribution Probability of firing: P F = ρ Δt 0 Δt? (i) Continuous time prob to survive (ii) Discrete time steps Δt 0
22 Neuronal Dynamics 5.3b. Inhomogeneous Poisson Process rate changes Probability of firing Survivor function Δt Interval distribution P = ρ() t Δt F S( t tˆ ) exp( ρ( t ') dt ') = t tˆ
23 Neuronal Dynamics 5.3b. Inhomogeneous Poisson Process t A(t) I(t) T n sp PSTH ( t) = n( t; t + Δt) K Δt n( t; t + Δt) A( t) = NΔt population activity inhomogeneous Poisson model consistent with rate coding
24 Neuronal Dynamics 5.3b. Inhomogeneous Poisson Process Probability of firing P = ρ() t Δt F Survivor function S( t tˆ ) exp( ρ( t ') dt ') = Interval distribution t tˆ P( t tˆ ) ρ( t)exp( ρ( t ') dt ') = t tˆ
25 Neuronal Dynamics Quiz 5.3. A Homogeneous Poisson Process: A spike train is generated by a homogeneous Poisson process with rate 25Hz with time steps of 0.1ms. [ ] The most likely interspike interval is 25ms. [ ] The most likely interspike interval is 40 ms. [ ] The most likely interspike interval is 0.1ms [ ] We can t say. B Inhomogeneous Poisson Process: A spike train is generated by an inhomogeneous Poisson process with a rate that oscillates periodically (sine wave) between 0 and 50Hz (mean 25Hz). A first spike has been fired at a time when the rate was at its maximum. Time steps are 0.1ms. [ ] The most likely interspike interval is 25ms. [ ] The most likely interspike interval is 40 ms. [ ] The most likely interspike interval is 0.1ms. [ ] We can t say.
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