DEIM Forum 2014 D5-2

Measuring spike train distance from multichannel spike trains data simulated
by coupled escape rate modelWei HU†

TaroTEZUKA‡†Graduate School of Library, Information and Media Studies, University of Tsukuba
1-2 Kasuga, Tsukuba, Ibaraki 305-8577 Japan‡Faculty of Library, Information and Media Science, University of Tsukuba
1-2 Kasuga, Tsukuba, Ibaraki 305-8577 Japan
E-mail:Abstract†hw812317948@gmail.com,‡tesuka@slis.tsukuba.co.jp

Estimating the population activity patterns between two or more spike trains is a fundamental problem in studying

neural coding in computational neuroscience. In recent years, there are many different methods proposed to build a framework
to deal with these problems by using spike train metric. Here we suggest a kernel method for multichannel spike trains that canprovide an opportunity to measure spike trains. As kernels can be used for various tasks in machine learning, including
regression, clustering and dimension reduction. We believe this method is effective at measuring multichannel spike trains
simulated using a distance.
Keyword

spike train distance, coupled escape rate model, kernel methods, multichannel spike trains, neuronal coding

1. Introduction

work that uses distances to evaluate similarity of spiketrains. The aim of this paper is to propose kernels for

In our brain, there are over 100 billion neurons, and they

spike trains. Specifically we use coupled escape rate

communicate with each other using pulses by generating

model (CERM) [2] to simulate spike trains data, and apply

characteristic electrical signals called action potentials or

the memoryless cross intensity kernel, which is calculated

spikes. We don’t know how these neurons transmit

by convolving the spike trains and a smoothing function.

information rapidly and what relationship they have

We then obtain a distance between spike trains [3][4][5].

between neural response and stimulus attributes.

The remainder of this paper is organized as follows. TheWe not only want to find the structure of neuron

second section describes related work about spike train

networks, but also want to know information contents ofdistances and spike train kernels. In the third and fourth

neural signals system. It is therefore important to know

section we will describe the memoryless cross intensity

how information is coded by neurons. To this end manyfor multichannel spike train model and discuss the coupledresearchers are working on the analysis of spike trains.

escape rate model to simulate multichannel spike trains.

A spike train is a series of spikes or action potentials
fired by a neuron. Although there is a method toAnd then, a distance is used to evaluate them. Finally, wewill give the conclusion.

electrically visualize action potential propagation andnetwork topology in cortical neurons [1], it can’t explain2.Related work

how spike works and the temporal structure of spike trains.
There is a big motivation to understand how to analyze

A way to construct a mathematical framework for spikeand decode the information expressed by spike trains in

train is the basic idea to define a distance between spikecomputational neuroscience. Recently, there are sometrains in a spike metric space [6]. Various distances havepublicly available spike train data. Also there is somebeen proposed for measuring the similarity between spike𝑘 𝑥! , 𝑦!     𝑘 𝑥! , 𝑦!         ⋯ 𝑘 𝑥! , 𝑦!
𝑘 𝑥! , 𝑦!     𝑘 𝑥! , 𝑦!
⋯ 𝑘 𝑥! , 𝑦!
𝑘(x, y) =⋮⋱⋮
𝑘 𝑥! , 𝑦!     𝑘(𝑥! , 𝑦! ) ⋯ 𝑘(𝑥! , 𝑦! )trains, including the Victor-Purpura distance [6], vanRossum distance [6], ISI (Inter Spike Interval) distance [7]and spike distance [8].

(2)

The Victor-Purpura metric [9][10] and the van Rossum
metric [9][10] both compare two different responses fromIf x = y, we will get a diagonal matrix.

the same neurons corresponding to different trials.

4. Method
3. ModelIt is well known that the firing rate is changed over time in
3.1. Memoryless cross intensity for spike trainsneural code experiments. We use inhomogeneous Poissonprocess to simulate spike trains. We also use the coupled

There are many studies in the literature to use innerescape rate model (CERM) [2] to model interactions

products to solve general machine learning problems. Hereamong neurons. It takes many parameters from a realistic

we use two spike train metrics to define inner products onnetwork model.

functional representations of spike trains [3][4][5].

Here we considered a case in which there are twoConsider two spike trains, x, y, x=(x 1 , x 2 , ... , x i ), y=(y 1 ,y 2 , ... , y j ), with i, j  ∈ N, and represent the spike train as asum of Dirac delta functions 𝑥(𝑡) =! δ(t − t ! ) ,rate model is defined as follows.

thendefine the intensity function by convolving the spike train
and a smoothing function h, as in λ   (t) =neurons a, b with synaptic coupling. The coupled escape        𝜆! 𝑡 = exp  [  𝑢! +    𝛼! 𝑥! 𝑡    +    𝐽!" (𝑡)𝑠!" (𝑡)  ](3)

𝜆! 𝑡 = exp  [  𝑢! +    𝛼! 𝑥! 𝑡    +    𝐽!" (𝑡)𝑠!" (𝑡)  ](4)! h(t − t ! ). Here,t i is the timing of the ith spike. The inner product can be
simply defined as!!!

k(x, y   ) =λ!   (t)λ!   (t) dt

This is the memoryless cross intensity kernel (mCI kernel)    !

=    −   !      +   𝑘 𝛿  (  𝑡 −    𝑡𝑎,𝑘 )	  	  (5)𝑑𝑠𝑎𝑏𝑠=    −   τ𝑎𝑏      +   𝑑𝑡𝑠𝑘 𝛿  (  𝑡 −    𝑡𝑏,𝑘 )  (6)!"(1)!!

proposed by Antonio Paiva [3]. In our experiments, we

λ i (t) is the instantaneous firing rate of neuron i, mostused a Gaussian function for h, which is a very common

parameters are indicated in Figure 1. t i,k is the k-th spikechoice in the analysis of spike trains.

time of neuron i, and δ(t) is the Dirac delta function. Boththe time constants τ m and τ s were 10 ms. (5) and (6) were

3 . 2 . Memoryless cross intensity for multichannel spike

calculated with a time step of 1 ms.

trainsOur aim in this paper is to extend the mCI kernel tomultichannel spike trains, and evaluate it using simulated
data.
Consider a case where there is a pair of spike train
metrics x, y. Let x=(x 1 , x 2 , ... , x i ) and y=(y 1 , y 2 , ... , y j )where each x i and y j indicates a spike trains and i, j  ∈ N.
Here, x i will be called a component of x.
The most general way to define a kernel is to use thetwo spike train metrics x and y directly as variables
without imposing any structure. As before, a generalkernel can be expressed as k (x, y), here we can define theFigure 1 CERM model parameters: {u a , b , a a , b , J a b , b a , x a , b (t),kernel on a pair of their components x i and y j as k(x i , y j ).

s a b , b a (t) }.

The memoryless cross intensity for multichannel spike

5. Evaluation

trains can be written as,
In this section, we first use the coupled escape rate modelto get the simulation spike trains data. We also compare

other means of analyzing data.

the average instantaneous firing rate by changing one ofparameters in the coupled escape rate model. Specifically,
The external input to one neuron is changed. Finally, weshow how a distance could be used to evaluate themultichannel mCI in a rigorous manner.
5.1. Simulation data
We used the coupled escape rate model defined in theprevious section. There are two neurons used in thesimulation

thatproduce

Poissonspike

trainswith

time-varying firing rates.
Most of the parameters were summarized in Table 1.
The data had been classified into ten conditions byFigure 1.

changing u 1 =[1.4: 0.2: 3.2]. For each condition, 20 trials

conditions with each condition carried out for 20 times. We canwere carried out. The time step is set to 1 ms and each trial

see a subtle difference for each condition by using the average

lasted for 500ms. Our simulation model was run using

firing rate of multichannel spike trains.

When changing u 1 =[1.4: 0.2: 3.2], we will get ten

MATLAB.
The norm distance is a commonly used distance
Table 1 Summary of parameters used in our simulationobtained from a kernel [5]. The norm distance between

model.

two spike trains is defined as follows.

a1,a2

J 12 ,J 21

u1-0.6, -0.9

𝑑 𝑡, 𝑠 =𝑘 𝑡, 𝑡 − 2𝑘 𝑡, 𝑠 + 𝑘(𝑠, 𝑠)(7)-0.5, -0.4
[1.4 : 0.2 : 3.2]

u2

1.7

x 1 (1), x 2 (1)0, 0s 12 (1), s 21 (1)0, 0With the same stimuli, the neuron will get different
respond which one maybe has subtle differences. During
each condition, we just change the external input toneuron 1, and get 200 times spike trains data in total. Sowhen u 1 = 1.4, we can get 20 times spike trains as t=(t 1 , t 2 ,t 3 , ... , t 20 ), the same to when u 1 = 1.6, another 20 times
spike trains as s=(s 1 , s 2 , s 3 , ... , s 20 ) and so on.
As the result of the of simulation model, in Figure 1 weFigure 2.

We calculate the norm distance for ten times, one iscompute the average firing rate of multichannel spikefor the base condition D m C I (t u 1 ,t u 1 ), and others are between thetrains.

base condition and each other condition D m C I (t u 1 ,t u x ), x =2,3, ... ,10.

5.2. Evaluation multichannel mCI
As this experiment has proposed for ten conditions. InSince we get multichannel spike trains t and s using theorder to see the obvious difference, we set u 1 = 1.4, t=(t 1 ,coupled escape rate model, we should evaluate them byt 2 , t 3 , ... , t 20 ), as the base condition to compare to otherchanging it to a distance. This is because a distance isconditions, s=(s 1 , s 2 , s 3 , ... , s 20 ).

usually used as a basis for classification, regression andFrom Figure 2, we can see a result that the distance of D t,s9 times is larger than the distance of D t,t . This means thatwith the same u 1 , the distance between spike trains (t i , t j )is small and with the different u 1 , the distance betweenicalculate the norm distance for ten times.

For the condition of J 12 =[-0.5: 0.05: -0.05], we also
can the same result from Figure 5 and Figure 6.

jspike trains (t , t ) is much smaller than the distance
between spike trains (t i , s j ).
5.3. Comparison with other parameters
Since we get the result by changing the parameter of u 1 ,it’s necessary to prove this conclusion in the other way. So
we do the same experiment with changing the parameter of
J 12 and a 1 .
In the evaluation with a 1 =[-0.6: 0.05: -0.15], we usedtheir original parameter settings which obtained u 1 =1.4.
The result is showed in Figure 3 and Figure 4. AS there isa big change with the average firing rate, the difference ofthe norm distance is large.

Figure 5.

When changing J 1 2 =[-0.5: 0.05: -0.05], the averagefiring rate of multichannel spike trains.

Figure 3.

When changing a 1 =[-0.6: 0.05: -0.15], the averagefiring rate of multichannel spike trains.

Figure 6.

We consider J 1 2 = -0.5 as the base condition andcalculate the norm distance for ten times.

6. Conclusion
In order to analyze multichannel spike trains, we proposeda new kernel that extends memoryless cross intensity
kernel. We evaluated it using simulated spike trains
generated from the coupled escape rate model. The result
showed that this method is effective at measuring

Figure 4.

We consider a 1 = -0.6 as the base condition andmultichannel spike trains.
In future work, we plan to extend this linear
functional kernel to nonlinear functional kernel toevaluate their performance. We also plan to use othersimulation models. They will make simulated datathat more similar to the real spike trains.

Acknowledgements[10] Conor Houghton and Thomas Kreuz, “Measures ofspike train synchrony: From single neurons toThis work was supported in part by JSPS KAKENHI Grant

populations”, In Multiscale analysis and nonlinear

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Pesenson and Heinz Georg Schuster, Wiley, 2013.

7.

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