The Computational Structure of Spike Trains
Robert Heinz
1,2
Haslinger
Cosma Rohilla
3
Shalizi
1 Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown MA 2 Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge MA
3 Department of Statistics, Carnegie Mellon University, Pittsburgh, PA
In an experiment we measure spikes
generated
by
some
unknown
dynamical process.
CSMs represent this process as a
directed graph in which the nodes
specify the dynamical systems states {S}.
The directed edges correspond to
possible transitions between the states
and are labled with the symbol (spike or
no spike) emitted during the transition.
p=0.5 Bernoulli Spikes with 1 msec
Refractory Period
0 1
0 0 1
1
0
RAT BARREL CORTEX DATA
CSMs are minimal, optimally predictive hidden Markov models (HMM) and
constitute a “program” which captures the statistical structure of the spike train.
All CSMs were constructed by applying the CSSR algorithm and ISI
bootstrapping to 200 seconds of simulated data with 1 millisecond time
steps (N=200,000). No assumptions as to model form were made.
CSMs were constructed from spike data recorded by tetrodes in alpha
chloralose anesthetized rat primary somatosensory SI (barrel) cortex.
CSMs were generated for both spontaneous spiking and stimulus
evoked (by vibrissa deflection) spiking.
(1) 40 Hz Bernoulli Process
(3) 40 Hz Bernoulli Process, “Soft” Refractory Period
(5) 90 Seconds of Spontaneous Spiking
0 | 0.96
A
1 | 0.04
1 0
Random spiking (40 Hz, p=0.04 spikes/msec
IID) has no structure. The state of the spike
generative process is always the same,
regardless of the past spiking history.
History Dependent Firing Rate
0.1
0.06
0.04
A
0.02
0
0
5
10
15
20
Time Since Spike (msec)
25
0 | 0.5
1 | 0.5
0|1
A
0 1 0 0 0 0 0
B
Pr(A) = 2/3
Pr(B) = 1/3
The Causal State Splitting Reconstruction (CSSR) algorithm is a non-parametric
method for generating CSMs. No assumptions about the output structure are
made.
Using CSM’s we objectively quantify the number of bits needed to describe both
the complexity (structure) and randomness of both simulated and
experimentally recorded spike trains.
0 | 0.96
B
1 | 0.04
1 0 0 0 0 0 0 0 1
0|1
C
0|1
D
0|1
1 | 0.003
0 | 1.000
1 | 0.039
0 | 0.928
0|1
A
1 | 0.069
1 | 0.067
0|1
0 | 0.947
0 | 0.947
1 | 0.041
0 | 0.953
CSMs complement traditional spike train analyses by describing the
computational structure of spike trains, information which has previously
been unavailable.
J
0 | 0.968
0 | 0.924
1 | 0.032
K
K
0 | 0.921
0 | 0.959
L
L
0 | 0.925
C = H[ St ] = −E[log(Pr( St ))]
0 | 0.964
M
M
0 | 0.920
0 | 0.958
N
N
0 | 0.922
O
Pr( X
|X
future history
= x) = Pr( X
∞
t +1
|X
t
−∞
= y)
St = ε ( X −t ∞ = x) =
{ y : Pr( X t∞+1 | X −t ∞ = y ) = Pr( X t∞+1 | X −t ∞ = x)}
Causal States are Minimal and Predictively Sufficient
Causal states maximize the mutual information between the set of all possible
predictive statistics and the future history. I[ X t∞+1; St ] = I[ X t∞+1; X −t ∞ ]
CSMs are sufficient because they capture all of the probablistic dependencies
present in the spike train and they are minimal because they are constructed to
be the most compact representation which does so.
CSMs contain within them the history dependent instantaneous spike rate, e.g
the probablility of emitting a “1” given the current state. λ (t + 1 | H t ) = Pr( xt +1 = 1 | St )
0.04
The Algorithmic Information “K” of a string is the length of the shortest
program which reproduces the string exactly and then stops. For a
stochastic string this coincides with the Shannon Entropy.
0.02
5
10
15
20
25
30
msec
35
40
The only input (other than the data) is a History Length “L” which is chosen by
minimizing the Baysian Information Criterion and bootstrapping 99% confidence
bounds on the ISI distribution from the CSM.
5
10
15
5
0 .1
5
10
15
20
25
30
35
40
10
15
0 | 0.96
A
1 | 0.04
0
10
20
msec
30
40
50
30
spikes/msec
A
B
0 | 0.957
R = H[ X t +1 | St , St +1 ] = −E[log(Pr( X t +1| | St , St +1 ))]
J
1 | 0.06
0 | 0.816
1 | 0.122
1 | 0.05
K
0 | 0.677
1 | 0.265
1 | 0.366
1 | 0.03
0 | 0.634
1 | 0.02
0 | 0.735
X
P
spikes/msec
Time Dependent Firing Rate
λCSM(t)
λPSTH(t)
λmodel(t)
0.6
0.4
0.2
0.02
0
0
5
10
15
10
5
10
15
20
25
msec
30
35
40
45
50
16 States
C= 0.89 J=0.27 R=0.0007
30
The amount of information supplied to the spike generation process by
external influences can be extracted from the CSM using the Kullback Leibler
Divergence between the true spike distribution and that predicted by the
CSM. This can be calculated using the entropy rate.
20
25
30
Complexity C(t)
0 | 0.98
V
1 | 0.02
35
40
50
5
10
15
3
20
25
30
Entropies
35
1
5
10
15
20
25
30
45
50
J(t)
R(t)
H(t)
2
0
0
40
35
Time since stimulus (msec)
40
50
0 | 0.96
1 | 0.05
Model Neuron
1 | 0.05
ZZ
1 | 0.01
1 | 0.04
0.02
25
msec
30
35
40
45
0.5
0.015
0.01
0.005
5
10
15
20
25
30
35
Time since stimulus (msec)
40
45
50
0
0
5
10
15
20
25
30
35
40
time since stimulus presentation (msec)
45
50
0 | 0.99
C2
Time Dependent Firing Rate
0
0
5
10
15
5
10
15
3
20
0.02
50
The Authors acknowledge David Feldman, Kristina Klinkner Rob Kass, Emery
Brown and Christopher Moore for thoughtful discussions.
λCSM(t)
λPSTH(t)
20
25
30
35
40
45
50
20
25
30
35
40
45
50
Complexity C(t)
2.5
2
1.5
0
29 States
ACKNOWLEDGEMENTS
C1
0.04
0.04
15
Stimulus Driven Entropy (ΔH(t))
0.025
1
0
0
0.02
10
Barrel Cortex Neuron
Stimulus Driven Entropy (ΔH(t))
1.5
0 | 0.99
C=1.97 J=0.11 R=0.004
45
1 | 0.05
1 | 0.03
1 | 0.03
1 | 0.03
1 | 0.03
1 | 0.02
ISI Distribution
5
∆ H = H CSM (t) − H true (t)
λ true (t)
1 − λ true (t)
+ (1 − λ true (t)) log
= λ true (t) log
λ CSM (t)
1 − λ CSM (t)
1 | 0.03
1 | 0.02
0
0
The difference between the CSM’s entropy rate and the true entropy
rate is the amount of information supplied by external processes
1 | 0.04
0.06
45
H true (t) = − λ true (t) log [λ true (t)] − (1 − λ true (t)) log [1 − λ true (t )]
1 | 0.06
1 | 0.02
0 | 0.98
H CSM (t) + log [1 − λ CSM (t)]
log [1− λ CSM (t)] − log [λ CSM (t)]
and the true entropy rate as a function of time since stimulus presentation
1 | 0.04
S
W
0.08
5
0
0
λ true (t) =
1 | 0.04
0 | 0.97
0 | 0.98
O
0
0
25
We can calculate the true time dependent firing probability
0 | 0.977
P
0 | 0.97
N
0.04
20
H CSM (t) = − λ true (t) log [λ CSM (t)] − (1 − λ true (t)) log [1 − λ CSM (t )]
1 | 0.03
T
0 | 0.98
U
0.08
15
H2
0 | 0.97
R
1 | 0.03
1 | 0.184
0.1
10
1 | 0.01 0 | 0.99
0 | 0.97
Q
0 | 0.608
0.12
5
time since most recent spike (msec)
If we calculate the entropy over a stimulus ensemble as a function
of the time since stimulus presentation:
0 | 0.96
1 | 0.04
J
ISI Distribution
J(t)
R(t)
H(t)
H
1 | 0.04
0 | 0.96
0 | 0.839
Because the states are Markovian, the complexity only needs to be described
once, while the internal entropy rate and residual randomness are updated at
each time step. The algorithmic information of the spike train is:
CSMs are constructed assuming that the past spiking is the only available
information for predicting the future spiking. This can lead to inaccurate
firing rates when the spiking process is being externally stimulated.
H1
O
0 | 0.795
1 | 0.392
Entropies
0.2
K
0 | 0.97
L
0 | 0.97
M
0 | 0.96
0 | 0.97 N
0 | 0.98
0 | 0.791
M
30
E
0 | 0.94
F
0 | 0.95
G
I
0 | 0.821
0 | 0.878
1 | 0.323
25
0 | 0.95
0 | 0.96
H
1 | 0.209
20
2
0 | 0.95
I
Residual Randomness
randomness
C
0 | 0.95
0 | 0.945
1 | 0.161
15
4
0 | 0.99
D
1 | 0.179
J = H[ St +1 | St ] = −E[log(Pr( St +1 | St ))]
10
6
1 | 0.01
1 | 0.01
G
1 | 0.118
5
Complexity C(t)
0 | 0.99
1 | 0.052
1 | 0.115
30
0.4
0 | 0.948
1 | 0.205
structure
0
25
F
Internal Entropy Rate
= C + N( J + R )
0.01
20
0 | 0.882
C = H[ St ] = −E[log(Pr( St ))]
25
8
50
0
0
D
Complexity
20
0 | 0.94
1 | 0.055
E
15
(6) 300 Seconds of Periodic Vibrissae Deflections
J(t)
H(t)
C
0 | 0.885
10
30
0 | 0.944
0.06
0.02
45
B
=C + J +R
Ε [ K ( x )] = H[ St ] + N{H[ St +1 | St ] + H[ X t +1 | St , St +1 ]}
40
1 | 0.043
0 .2
+ H[ X t +1 | St , St +1 ]
n
1
0.03
25
time since most recent spike (msec)
A
bits
0.04
msec
35
C=1.02 J=0.10 R=0.005
30
R(t)
0
0
ISI Distribution: Bernoulli Model
0.05
20
Entropies
0.2
bits
CSM Generated from 200 sec
of a 40 Hz Bernoulli Process
30
5
2
0 .3
0
25
0
0
0.4
H[ St , St +1 , X t +1 ] = H[ St ] + H[ St +1 | St ]
Complexity Only Needs to be Described Once!
25
20
4
L
CSSR is a non-parametric method and makes no assumptions as to the form of the
CSM.
20
Complexity (C(t))
0 .4
1 | 0.056
CSMs can be generated using the Causal State Splitting Reconstruction
(CSSR) algorithm freely downloadable from http://bactra.org/CSSR/
50
15
0 .5
At time “t” the spike generating process is described by 1) the current state, 2)
the next state, 3) the next emitted symbol. This suggests a natural
decomposition of the entropy.
Residual Randomness: Randomness
in the symbol emission which is not
described by the CSM .
10
Stimulus Driven Firing Rate
0 .6
Decomposition into Structure and Randomness
CAUSAL STATE MODELS ARE
GENERATED FROM DATA
5
0
0
Time Since Stimulus (msec)
Internal Entropy Rate: The average
number of bits per time step, beyond
the complexity, needed to describe
the exact state sequence.
0
0
λmodel(t)
15
14 States
(4) “Stimulated” Inhomogeneous Bernoulli
0
Complexity: Amount of information
(in bits) about the system’s past
relevant for predicting its future.
45
C=3.16 J=0.25 R=0
Ε [ K ( x1 )] = H ( x1 ) + o(n) = H[T ] + H[ x1| T ]
structure
randomness
where T is a minimal sufficient statistic which describes the structure
the sequence of causal states S0n is a minimal sufficient statistic
0.05
17 States
Given a symbol string x1n, K ( x1n ) can be written
as:
n
n
n
History Dependent Firing rate
λCSM(t)
λPSTH(t)
0.1
6
0
0
10
0
0
bits
ISI Distribution
0.08
5
0.02
spikes/msec
X
∞
t +1
x, y { X } are equivalent if
t
−∞
0
0
0.06
t
−∞
∞
t +1
Q
THE INFLUENCE OF STIMULI
λCSM(t)
λPSTH(t)
0.04
bits
1 0
0.02
0.06
bits
past history
0
0 | 0.931
bits
X −t ∞
0 1
t
0.04
History Dependent Firing Rate
0.4
bits
These equivalence classes are the
Causal States.
0 0 1
0.06
bits
Two past histories “x” and “y” are
equivalent when they have the same
conditional distribution of future
histories.
1
0.08
P
spikes/msec
A dynamical system’s “state” consists of
the information relevant for predicting
its future evolution.
0 | 0.926
ALGORITHMIC INFORMATION
AND RANDOMNESS
ISI Distribution
0.1
spikes/msec
WHAT ARE CAUSAL STATES?
Possible applications include the study of the structural complexity of
different cortical states, or comparisons of the complexity of neural activity
during different sensory, motor or cognitive tasks.
I
J
The structure can be sumarized by the COMPLEXITY
0 | 0.959
0 | 0.948
0 | 0.933
0 | 0.931
1 | 0.042
H
1 | 0.074
I
1 | 0.036
G
1 | 0.041
1 | 0.078
A lower bound on the amount of information supplied to the spike
generation process by external influences (stimuli) can be derived from the
CSM.
1 | 0.047
0 | 0.954
1 | 0.046
1 | 0.075
H
F
F
1 | 0.079
0 | 0.961
1 | 0.052
0 | 0.935
1 | 0.076
0 | 0.976
1 | 0.053
E
1 | 0.069
F
The CSSR algorithm was used to generate CSMs from both simulated and
experimentally recorded neural spike trains without making any
assumptions as to the form of the CSM.
1 | 0.065
1 | 0.053
0 | 0.988
1 | 0.080
0 | 0.961
D
G
E
1 | 0.039
C
1 | 0.024
E
0 0
0 | 0.991
1 | 0.012
0 | 0.997
The states of the CSM are equivalence classes of past spiking histories. Each
member of a class has the same distribution of future spiking histories
1 | 0.010
B
1 | 0.072
D
Imposing a 5 msec refractory period after each spike imposes statistical
structure on the spike train. Six states are required to describe the structure.
The Causal State Splitting Reconstruction Algorithm can be used to
automatically determine the computational structure of spike trains in the
form of a minimal optimally predictive hidden Markov model or Causal
State Model (CSM).
A
1 | 0.009
B
C
(2) 40 Hz Bernoulli Process, 5 ms Refractory Period
0 | 0.957
1 | 0.043
30
0 | 1.000
No a priori
Assumptions
Neurons are complicated biophysical devices, which may not perform
computation in a simple manner.
0 | 0.990
spike
0.08
DISCUSSION
bits
We quantify the computational structure of a spike train in the form of an
optimally predictive hidden Markov model or Causal State Model solely though
analysis of the output spikes.
SIMULATED DATA
bits
Neurons are computational devices, transforming synaptic input (hard to
measure) into output spikes (routinely measured).
CAUSAL STATE MODELS
QUANTIFY STRUCTURE
Spikes/msec
INTRODUCTION
Entropies
J(t)
R(t)
H(t)
0.2
0
0
5
10
15
20
25
30
35
40
45
time since stimulus presentation (msec)
50
The Authors thank Mark Andermann and Christopher Moore for use of their
rat barrel cortex data.
This work was funded in part by NIH K25 NS052422