CA1 Region of Rat Hippocampus
Spike Time Series
Time (sec)
Position Specific Spiking
Inhomogeneous Rate Model: Gaussian Place Field
⎧1 ′-1 ⎫ λ(t)=exp α- (x(t)-μ)W (x(t)-μ)
Model covariate: position
x(t) = (x1(t),x2 (t)) Model parameters:
maximum field height
scale matrix
e x p {α } center
⎡ σ 12
⎢σ σ2⎥
μ = (μ1 ,μ2 )
⎢⎥ ⎣12 2⎦
⎨2⎬ ⎩⎭
W=
σ 1 2 ⎤
Point Process Data Likelihood
IntensityModel: λk =λ(x(tk)|Hk) ObservedSpikeData:ΔNk =N(tk +Δt)−N(tk) Data Likelihood:
T
L(Spike Train |θ) = exp(∑log(λ Δt)ΔN −λ Δt)
Estimate model parameters by maximum likelihood
k=1
kkk
ˆ L(μ ,μ ;α =α,W =W)
12
ˆ
ˆ L(μ ,μ ;α =α,W =W)
12
ˆ
ˆ L(μ ,μ ;α =α,W =W)
12
ˆ
ˆ L(μ ,μ ;α =α,W =W)
12
ˆ
ˆ L(μ;α=α,μ =μ,W=W)
ˆ L(μ2;α=α,μ1 =μ1,W=W)
1 ˆ2ˆ2
ˆ ˆ
μ1
μ2
Likelihood Slices
Parameter Estimates
⎧1 ′-1 ⎫ λ(t)=exp α- (x(t)-μ)W (x(t)-μ)
⎨2⎬ ⎩⎭
ˆ
e x p {α } = 6 . 8 2 ± 0 . 4 8 H z
μˆ1 = −0.12 ± 0.09 μˆ2 = −0.32 ± 0.11
W = ⎡0.072 0 ⎤ ± ⎡0.001
0.004⎤
⎢ 0 0.111⎥ ⎣⎦⎣⎦
⎢0.004
0.003⎥
Model Fit
Model Fit
Spiking Rate in Time
Time (sec)
4 2 0
# Spikes/33 msec bin
Firing Rate (Hz)
Goodness-of-Fit Questions
• How well does this model describe the data?
• How does this model compare to others? • How can we refine this model?
Linear exponential model:
Gaussian shaped model:
Three Models
λ(t)=exp α-βx(t) {}
⎧1 ′-1 ⎫ λ(t)=exp α- (x(t)-μ)W (x(t)-μ)
⎨2⎬ ⎩⎭
Gaussian model with random signal dependence:
⎧1′-1 ⎫ λ(t)=exp α- (x(t)-μ)W (x(t)-μ)+γs(t)
⎨2⎬ ⎩⎭
Linear
Est Gaus 1.48 ±.12 eα
Est -6.82±.48
G+R Est
eα -6.78±.51
eα β1 β2
-.49±.08 μ1 1.21±.08 μ2
-.12±.09
μ1 -.14±.08 μ2 -.30±.10
w
-.32±.11 .072±.001 0.00±.004 .111±.003
w .072±.002 11
w
w -0.00±.006 12
Model MLEs
11
12 w22
w22 .110±.022 γ .063±.135
Linear exponential
Gaussian
Gaussian + Random signal
-2*Log Likelihood at MLE:
AIC:
16181
12555
12554
16187
12567
12568
Model Fits
Time Rescaling
ISI Histogram
Linear exponential Gaussian Gaussian + Random
KS Statistics:
KS Plots
Linear exponential Gaussian Gaussian + Random
0.315 0.113 0.095
Model QQ Plots
Linear exponential Gaussian Gaussian + Random
Rescaled ACFs
Linear exponential Gaussian Gaussian + Random
Point Process Residuals
Linear exponential
Gaussian
Residuals vs x2
Residuals vs x2
Residuals vs y2
Residuals vs y2
Lag
Lag
Sample Fano Factor
Sample Fano factor distribution for spikes binned at 33 ms
Sample Fano factor
Linear model:
Addition of Random Signal:
– Doesn’talterspatialfieldproperties – Decreasedpredictivequality
– Poorestexplanatory/predictivequality
– Missesbothsmallandlargequantiles
– Correlationsatdistantlags
– Quadraticcomponentmissingfrom model
Quadratic model:
– Betterexplanatory/predictivequality – Misseslargequantiles
– Correlationsatsmalllags
All models:
– Incompletespecifications(KStest) – Cannotdescribelargequantiles
– DatanotinhomogeneousPoisson
Summary