Neural Signal Multiplexing via Compressed Sensing

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by  Nithin Nagaraj

Neural Signal Multiplexing via Compressed Sensing
Nithin Nagaraj∗ and K. R. Sahasranand†
∗Consciousness Studies Programme, School of Humanities, National Institute of Advanced Studies,
Indian Institute of Science Campus, Bangalore, INDIA.
†Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, INDIA.
(Email: [email protected], [email protected])
Abstract—Transport of neural signals in the brain is chal- representation of an otherwise noisy and variable sensory
lenging, owing to neural interference and neural noise. There environment.
6 is experimental evidence of multiplexing of sensory information
In this paper, we propose a phenomenological model to
1 acrosspopulationofneurons,particularlyinthevertebratevisual
study the feasibility of such efficient multiplexing of neural
0 and olfactory systems. Recently, it has been discovered that in
2 lateral intraparietal cortex of the brain, decision signals are signalsinthepresenceofneuralinterferencefromaverylarge
multiplexedwithdecision-irrelevantvisualsignals.Furthermore, number of neurons. We wish to point out the possibility of
n
it is well known that several cortical neurons exhibit chaotic such mechanisms from a theoretical perspective while trying
a
spikingpatterns.Multiplexingofchaoticneuralsignalsandtheir
J to use reasonable models which simulate the behaviour of
successfuldemultiplexingintheneuronsamidstinterferenceand
3 noise, is difficult to explain. In this work, a novel compressed actual neurons. To the best of our knowledge, there are no
1 sensingmodel for efficientmultiplexingof chaotic neuralsignals mathematicalmodels in the neuroscienceliterature which can
constructed using the Hindmarsh-Rose spiking model is pro- explain a mechanism of simultaneous compression, reliable
T] posed. The signals are multiplexed from a pre-synaptic neuron multiplexing and robust demultiplexing of chaotic neural
s.I tc1oo0m4itpsirnetnsesereifgdehrisbneongusrinninoggistyepcohnsnetu-iqsryuanleasp.sitgicnanlseuarnodn,dienmuthlteipplerxeesdencuesinogf stoignbaelspe(rafmorimdsitng10s4ucnceosissyfulslyig.nWales)hwopheichtothdeembornasintrasteeevmias
c Index Terms—neural signal multiplexing,compressed sensing, theoreticalargumentsandsimulationresultsthatourproposed
[
chaotic signals. compressed sensing model has all the three benefits.

1 The paper is organizedas follows. We briefly review exist-
v I. INTRODUCTION ing methods of multiplexing of chaotic signals and point out
4
Thenumberofneuronsinthehumanbrainisapproximately their drawbacksin Section II. None of these existing methods
1
86 million [1] which is roughly half the number of stars in seem feasible to model multiplexing of chaotic neural signals
2
3 our Milky Way galaxy. Each of these neurons is connected in the brain. We propose a novel model of multiplexing (and
0 anywhere between 102 to 104 other neighbouring neurons on demultiplexing)ofchaoticneuralsignalsusingthe framework
1. an average. It remains a mystery to a large extent as to how ofcompressedsensinginSectionIII.Weconcludewithfuture
0 informationencodedinactionpotentials(spikes)istransported research directions in Section IV.
6 fromoneneuroninthe middleofa labyrinthof neuronsto its
1 immediateneighbouramidstacacophonyofinterferingneural II. MULTIPLEXINGOF CHAOTIC SIGNALS
:
v signals and neural noise. The brain not only employs very Chaos as a possible neuronal code was proposed by re-
i efficientcompressionofinformationwithlowconsumptionof searchers in neuroscience owing to enormous experimental
X
energy,butalso isknowntorobustlymultiplexneuralsignals. evidence of chaos in the nerve cells, neural assemblies, be-
r
a In a recent study involving adult rhesus monkeys, sin- haviouralpatternsas wellas higherbrainfunctions.Thereare
gle neuron spiking measurements in the lateral intraparietal several advantages of chaos and nonlinear response models
area (LIP) showed that decision signals are multiplexed with over stochastic ones, especially the possibility of control of
decision-irrelevant visual signals [2]. Furthermore, the same the former. Korn et al., in their excellent review article [5],
studyalsorevealedthatneuronsinLIPexhibitdiversedynam- trace the history of the search for chaos in the nervous
ics indicatinga broadrange of neuralcomputations.Friedrich system providing pointers to numerous experimental studies.
etal. [3]foundthatoscillatoryfield potentialsin thezebrafish Dynamical approach to brain operations and cognition has
olfactory bulb multiplex information about complimentary been vindicated and there is a large body of compelling
stimulus features by the same population of neurons. These evidence of chaos at all levels.
can be retrieved selectively by biologically plausible mecha- The problem that we wish to address is the ability of
nisms. Panzeri et al. [4] present a strong case for temporal neurons to multiplex chaotic signals. To this end, we briefly
multiplexing where neural responses at different timescales review known methods for multiplexing of chaotic signals.
encode different stimulus attributes. They argue that such Startingfrom1996,continuoustimechaoticsignalshavebeen
multiplexing increases encoding capacity of neural responses multiplexedusing the notion of dual synchronization[6]. The
and offers advantages over single response timescales, thus idea of dual synchronization proposed by Liu and Davis [7]
enabling the brain to arrive at an information-rich and stable is to use a scalar signal to simultaneously synchronize two
differentpairsofchaoticoscillators.Dualsynchronizationhas where n = 1,...,M. Equivalently, the above equation could
been implementedin electroniccircuits[8], butthis hasprob- be expressed in matrix form:
lems such as difficulty of extendingto more than two signals,
sensitivitytonoiseandtheneedtocarrytwoscalarsignalsfor w1
 
multiplexing more than two signals (but action potentials are w2
trreasnuslmtsiitnteadsaysncjuhsrtonaizsaintigolneesrcroalraorfs4ig%naals).reEpvoernted1%byoLfiunoainsde  y(...1) = s1 s2 ... sN  ..... , (2)
Davis [7]. Symbolic dynamics-based approach to multiplex  y(M)    . 
two pairs of low frequency chaotic electronic circuits that  .. 
 . 
produceRo¨ssler-likeoscillationswasdemonstratedbyBlakely  
 wN 
and Corron [9]. However, even a small synchronization error
leadstohugeerrorstherebymakingitunsuitable.Nagarajand which can be further written succinctly as:
Vaidya [10] proposed multiplexing of discrete chaotic signals
in the presence of noise using the idea of symbolic sequence y =Ax. (3)
invariance. While they demonstrate multiplexing up to 20
Some points to note:
chaoticsignals, their methodis notrobustto hugeamountsof
noiseandasthenumberofsignalstobemultiplexedincreases, 1) It is quite reasonable to assume linear processing of in-
the noise tolerance comes down quite rapidly. putsasintheaboveequation.Forexample,theolfactory
Even if we grant that some of the above methods could be bulblinearlyprocessesfluctuatingodorinputs. Guptaet
modified to work with multiple chaotic signals, it is not at al.[11]haveshownthatindividualmitral/tuftedcells(in
allobviouswhetherthese can be extendedto multiplexneural anesthetized rats) sum inputs linearly across odors and
signals from neuronal models, as it has never been attempted time.
before.Mostimportantly,these methodswouldfailto address 2) In this work,we assume a valueof N =104. It may be
the problem that we are interested in multiplexing a small thecasethattherearefewerthan104neuronsconnecting
number of neural signals amidst 104 c−ompeting noisy neural to Z (and E). We have assumed this huge number as a
signalsfromneighbouringneurons.Thesemethodshavenever worst-case scenario, but our methods work with lower
been tested for such large number of signals, but this is values of N as well.
precisely what the brain has to contend with. 3) We shall assume that only a small fraction (say 1.5%)
of the N neural signals si reaching Z are from
{ }
neighbouring neurons. These are chaotic outputs of
III. PROPOSED APPROACH
the Hindmarsh-Rose neuron model (described below)
but corrupted with small amounts of additive noise.
The problem addressed in this work is stated as follows.
Hence, this small set of signals (1.5% of the columns
Given a particular pre-synaptic neuron (Z) and its eastern
of matrix A) is modeled as signal-dominant with low
post-synaptic neighbour (E), both of these being connected
to 104 other neurons, how does neuron Z simultaneously noise (additive Gaussian noise2 (0,0.01)). The re-
N
≈ maining 98.5% of the neural signals are completely
transport 4 neural signals s1(n), s2(n), s3(n) and s4(n)
dominated by additive Gaussian noise and are modeled
(corresponding to North, North-West, South-West and South
as i.i.d Gaussian samples (0,1). Even though these
neighbours respectively) to neuron E? Here n stands for
N
vast number of columns of A are also neural signals,
discretetimeindex(weassumeasampledneuralsignal,butin
they are effectively modeled as Gaussian noise, since
whatfollows,ourmethodscanbeextendedtocontinuoustime
by the time these neural signals arrive at neuron Z (as
signalsaswell).TheneuronZ canonlytransmitasinglescalar
wellasE -rememberthatE isnearestneighbourofZ),
signal across its synapse to neuronE. It is to be remembered
they are dominated completely by noise with no signal
thatallneuralsignalsare alwayscorruptedwith someamount
of noise, which we shall assume to be additive Gaussian1. component whatsoever.
4) The weights wi Ni=1 are scalars (real-valued) and cor-
{ }
respondtoeachpre-synapticneuronofZ (theseweights
A. Neural Signal Multiplexing
could be thought of as contributing to either an in-
At neuron Z, up to a first order approximation, we shall hibitory or an excitatory response). These weights are
assume thatit formsa weighted additionof all its inputnoisy completely unknown to E. It is reasonable to assume
neural signals. This can be mathematically represented as: that only a few neural signals (say k) among the 1.5%
signal-dominant columns of A (in our study we have
N taken k = 4) need to be multiplexed from Z to E.
y(n)= wi si(n), (1) Hence, we shall assume that only this k sparse set of
X ·
i=1 weights are significant (greater than some threshold T),
1Morerealistic noisemodelsneedtobetestedinthefuture. 2LetN(µ,σ2)denoteaGaussiandistributionwithmeanµ&varianceσ2.
thoughtheactualnumberofsignificantweights(k)itself similar fashion, the remaining 146 neural signals si(n) 1i=505
{ }
is unknown at E3. are also obtainedfromthe same modelwith the same settings
of r and I but with randomly chosen initial conditions. As
Hindmarsh-Rose Neuron Model
explained previously, the rest of the columns of A (9850) are
The Hindmarsh-Rose neuron model [12] is a widely used modeledasi.i.dGaussian (0,1)entries.Thus,wenowhave
modelforbursting-spikingdynamicsofthemembranevoltage our ensemble of 104 neurNal signals and neural noise which
S(t) of a single neuron. The equations of the model (in makes up Equation (3). For particular chosen values of the
dimensionless form) are: weights wi 1i=041, we plot y(n) in Fig. 3 along with neural
{ }
interference and noise.
S˙ = P +3S2 S3 Q+I, (4)
− −
P˙ = 1 5S2 P, (5)
− − 2
8
Q˙ = r[Q 4(S+ )]. (6)
− − 5 1.5
In the above set of differential equations, P(t) and Q(t) 1
represent auxiliary variables corresponding to fast and slow
0.5
transport processes across the membrane respectively. The
0
control parameters of the model are the external current ap-
plied,I,andtheinternalstateoftheneuron,r.Weuseavalue −0.5
of r = 0.0021, as it yields dynamics which corresponds to a
−1
realistic description of the electrophysics of axons of certain
neurons[13]. Themodelis capableofdisplayingbothregular −1.50 10 20 30 40 50 60 70 80 90 100
spiking as well as bursting (chaotic) dynamics for suitable
regimes of the parameter I. The bursting behaviour exhibits 2
chaoswherethereisalternatingirregularspikingfollowedbya
1.5
non-oscillatoryphasewhichsuddenlyendsinaburstofspikes.
Inourstudy,wehavechosenI =3.28whichisinthechaotic 1
0.5
2 0
1.5 −0.5
1 −1
0.5 −1.50 10 20 30 40 50 60 70 80 90 100
0
2
−0.5
1.5
−1
1
−1.5
0 10 20 30 40 50 60 70 80 90 100
0.5
0
Fig. 1. Noiseless neural signal s(n) simulated using the Hindmarsh-Rose
neuronmodel(dotted). Thenoisysignals1(n)(solid)isobtainedbyadding −0.5
N(0,0.1)tos(n).Allsuchplotsinthispaperdepictsignalamplitudeagainst
discrete timeindexn. −1
−1.5
0 10 20 30 40 50 60 70 80 90 100
regime [14]. We have sampled S(t) in steps of 1 time unit
toyielddiscrete-timeneuralsignals(n)andsubsequentlyadd
i.i.d.Gaussiannoise (0,0.01)toyieldthefinalsignals1(n) Fig. 2. The three noisy neural signals to be multiplexed along with s1(n).
N Top:s2(n),Middle: s3(n),Bottom:s4(n).
as shown in Fig. 1. In Fig. 2, we plot the remaining three
(noisy) signals s2(n), s3(n) and s4(n) obtained in a similar
Under normal circumstances, Equation (3) is unsolvable,
fashionfromrandomlychoseninitialconditionsofthe neuron
unlessM N.Thiswouldimplythattheneuralsignalsneed
model (followed by sampling and addition of i.i.d. Gaussian
≥
to be observed for a longer duration (thereby making M at
noise (0,0.01)). These are all essentially chaotic in nature
(the adNded noise, in fact makes it stochastic). The goal is to least=104).However,practicalconstraintssuchasimmediate
responsetostimulusdemandsolvingEquation(3)withshorter
multiplexallthefoursignalsfromneuronZ toneuronE.Ina
duration signals, which leads to the under-determined case
3Itisthissparsityorrathercompressibility ofxthatcomestoourrescue. (M < N). Surprisingly, this admits a unique solution which
recoverx. However,in our problem,we are onlyinterestedin
−0.5
identifying the columns that contribute to the signal and not
−1
theexactamplitudes.IntheCSparlance,thisiscalledsupport
−1.5
recovery [22].
−2
−2.5 C. Neural Signal Demultiplexing: A Compressed Sensing
−3 Model
−3.5
Consider the neuron E which has received the signal
−4
y = Ax from its neighbour Z. At E, it is only reasonable
−4.5
to assume that the signal set A is not perfectly available.
−5
0 10 20 30 40 50 60 70 80 90 100 We consider the case where the neuron E has to decode
using a matrix Aˆ (instead of A) where noisy versions 1.5%
Fig. 3. Signal vs. Noise. The mixed scalar signal y(n) (computed using of the columns of A are present in Aˆ (these correspond to
Equation (1)) transported from the pre-synaptic neuron Z to post-synaptic
signalsfrom nearbyneurons,the signal-dominantset) and the
neuronEisshowninsolid.Neuralinterferenceandnoisefrom9996neurons
(excluding theeffect ofthe4signals tobemultiplexed) isshownindotted. rest 98.5% of the columns are composed of independently
sampled Gaussian noise (these correspond to signals from
far away neurons, the noise-dominant set). This could be
could be accurately determined via a recently developed mathematically represented as follows: Let ai,aˆi denote the
ith columnsofAandAˆrespectively.Considerasetofindices,
signal processing paradigm known as compressed sensing (to
corresponding to the signal-dominant columns of A and let
be discussed in the next section), provided the wis to be
them be labelled 1,2,...,150. Then,
recoveredformasparse-set(oracompressibleset),andcertain
conditions are satisfied by the matrix composed of the noisy
aˆi =ai+ǫi,
neural signals.
where ǫis are independent Gaussian noise vectors. Now, the
B. Compressed Sensing problemisthatofrecoveringxgivenAˆandyinsteadofAand
Compressedsensing(CS)isarecentsignaldetectionframe- y. However,thisis equivalentto decodingx fromy =Ax+ǫ
150
work which has received considerable attention in the signal where ǫ= i=1xiǫi which has variance, say η. This can be
P
processing community [15], [16]. The signal measurement achieved by solving the convex program
process is linear as in, the observed vector y = Ax. If the
number of non zero entries of x is less than or equal to k, min||x||l1 subject to ||Ax−y||l2 ≤η,
we say x is k-sparse. The idea behind CS is that, if the whichgivesthe sparsesolutionx∗.In fact,it isshownin [15]
received signal is sparse (in an appropriate domain), most that, under this setting,
of what is received is thrown away during compression or
pmreoacseusrseinmgen[1ts7](.asThaufsu,nitctiisonpoosfsitbhlee stoparresdituyceofthxe)naunmdbuesrinogf ||x−x∗||l2 ≤C1ǫ+C2.||x−√xkK||l1,
randomprojectionsit is possible to recoverthe entire(sparse) where C1, C2 are constants and xK is the same as x at k
signal exactly [18]. Further, it has been shown that perfect leading entries and the rest set to zero. Surprisingly, even in
recovery is possible in the presence of noise as well [15]. theabsenceof98.5%ofcolumns(andtheremainingcolumns
Let x RN be a k-sparse signal. Let A RM×N denote corrupt), l1 minimization is seen to perform well. This is
the meas∈urement matrix (N = 104). The∈columns of the evident from our simulation result depicted in Figure 4. It
matrix correspond to the signals (vector-valued)generated by shows that signals multiplexed and transported from Z are
each of the 104 neurons. The received vector y RM is a successfully demultiplexed at E despite having only a noisy,
∈
linear combination of columns of A where the coefficients partial version of the mixing matrix A.
form the vector x i.e.,y =Ax.
D. Simulation settings
In our problemof interest, this means that the signals from
different neurons (columns of A) are multiplexed together We have used the following settings for our simulations.
during transmission to the nearby neuron. The number of N = 104, M = 100, 150 signal-dominant neural signals
signals multiplexed is equal to the sparsity of x. from Hindmarsh-Rose model (I = 3.28, r = 0.0021 and
Decodingof x giveny can be carried out througha variety randomly chosen initial conditions) with additive Gaussian
of techniques like greedy approach (algorithms like Orthogo- noise (0,0.01), 9850 noise-dominant neural signals sim-
N
nal MatchingPursuit[19]) or optimizationapproach(l1 mini- ulated as additive Gaussian noise (0,1), k = 4 signals
N
mization[20]),providedthe measurementmatrixAsatisfiesa to be multiplexed with weights w1 = 1.0, w2 = 0.9,
propertycommonlyreferredtoasRestrictedIsometryProperty w3 =0.8 and w4 =0.7. The weights for the signal-dominant
(RIP)[21].Forexample,whentheentriesarealldrawnfroma columns wi ii==1550 are drawn from uniform random distri-
{ }
Gaussian distribution (say (0,1)) appropriately normalized, bution in (0,0.02), weights for the noise-dominant columns
thematrixsatisfiesRIP.InNthesecases,itispossibletoexactly wi ii==115041 are drawn from uniform random distribution in
{ }
theory demands that the number of measurements M obeys
2 N
M C klog [24] for good recovery of x.
k
≈ ·
1.5
IV. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS
1
The transport model between adjacent neurons in the brain
0.5
that we have proposed has some unique features. The mixing
0
matrix A at pre-synaptic neuron Z and the corresponding
−0.5 matrix Aˆ at the post-synaptic neuron E need not have the
samenumberofneuralsignalsin thesameorder(numberand
−1
position of columns can be different, and is most likely the
−1.50 10 20 30 40 50 60 70 80 90 100 case in the brain). As a next step, it will be interesting to
study the theoretical properties of the matrix A (and Aˆ) and
Fig. 4. Demultiplexing: Original signal w3·s3 (dotted) at neuron Z and the limits of such multiplexing. Experiments with real neural
reconstructed signal wˆ3 ·sˆ3 (bold) at neuron E obtained by means of l1 signals in the brain, to test the proposed modelare necessary.
minimization.Similarrecoverywasobtainedforwˆ1·sˆ1,wˆ2·sˆ2andwˆ4·sˆ4,
butnotshownhereduetospaceconstraints.
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