Hodgkin-Huxley model

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Numerial and Computational Analysis of models for

Stohasti ativity of neurons

Ana Cláudiados Reis Valentim 1

LaboratórioNaionaldeComputaçãoCientía,LNCC,Petrópolis,RJ

Alexandre LoureiroMadureira 2

LaboratórioNaionaldeComputaçãoCientía,LNCC,Petrópolis,RJ

Hugode laCruz 3

FundaçãoGetúlioVargas,FGV,RiodeJaneiro,RJ

Abstrat. AurrentissueinComputationalNeurosieneistodevelopmodelsthatdesribe

theneuronalringauratelyandwithlowomputationalost. TheHodgkin-Huxleymodel

fullls the rstriteria but fails in the seond one. In order to handle it, there are other

simplermodels,as theIntegrate-and-re. Bothmethods presentdeterministi approahes,

and inorderto obtainmoreaurateresults,wehaveonsideredtheadditionofBrownian

noise. Wehaveobtainedmoreaurateresultsandomparedthem.

Key-words. Neurosiene. Numerial Methods. Hodgkin-Huxley model. Integrate-and-

Fire model. StohastiDierentialEquations.

1 Introdution

In1952,inthepaperAquantitativedesriptionofMembraneCurrentanditsappliation

toondutionandexitationinnerve[7℄,HodgkinandHuxleypresentedavoltage-dependent

modelforneuronalringasanonlinearphenomenon. Thismodelreeivedtheirnamesand

itisveryuselfulbeauseitapturesbiologialaspets,butitisdiulttosolveandpresents

very sensitive parameters [12℄. Simpler and heaper models were developed posteriorly,

aimingtoobtain,e.g.,ringrates [10℄. Consideringsimpliityandlowomputationalost,

theIntegrate-and-Fire (IF)modelis themost eient amongother models [9℄.

Bothmodelsapproahesneuronalringdeterministially. Thatisnotaurate,beause

it's known that, e.g., ortial and spontanous ativity ells do not present this kind of

behavior[2,8℄. Theseirregularitiesarebiologiallyausedbymanyfatorsandwerepresent

thesum ofthese irregularities inthemodeladding whitenoise onit [11℄.

We aim to obtain more aurate results for spike trains. We also aim to ompare HH

and IFmodels inrelation to omputational ostand outputprobabilitydistribution.

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2 The Hodgkin-Huxley model

TheHHmodeldesribesationpotenialsthroughananalogybetweentheellmembrane

and a simple eletri iruit [5℄. It is the most omplete model so far. It onsiders a

seletivelypermeableellmembrane,ontainingproteinhannelsforSodiumandPotassium

ions. Theopeningandlosingproess ofthesehannels ours stohastially,i.e.,onean

not estimatedeterministiallyifthe hannels will beopen orlosed inaertaininstant of

time.

The mathematial model onsists of a paraboli PDE and three ODEs. The rst

equation oftheHH modelshows howvoltage variesintimeand spae:

C ∂V

∂t = µ ∂ ² V

∂x ² −

3

X

i=1

g i (V − E i ) + I ext + R 1 ;

The voltage rate, whihis multipliedbythemembranespei apaitane C,is thesum

of the ioni urrents, the spatial term and input urrent

I _ext

^disturbed ^by ^an ^additive

noise

R 1

^. ^These îoni ûrrents âre ^given ^by ^Kirhho's ^law, êxpressed âs ôndutane

timesthe onentration gradient for eahion(voltageminus theNernstpotential). Aswe

have Sodium, Potassium andand leak hannels, theindex

i

ôrresponds ^to êah ôf^these,

respetively. The spatial variation is multiplied bya oeient of diusion

µ

^, responsible for the spatial diusion.

The other threeODEs modelhowthedimensionless gating variables vary intime. These

variablesrepresenttheprobabilityofopeningandlosingofativationgates,forPotassium

ion(

m

^),ând ôf âtivâtion ôr inativation gates,for Sodiumion(

h

^and

n

^).

dm

dt = (1 − m)α m (V ) − mβ m (V );

dh

dt = (1 − h)α h (V ) − hβ _h (V );

dn

dt = (1 − n)α _n (V ) − nβ _n (V ).

Thesevariations aregiven bythe onvexombination ofthegating variablebythegating

funtions, whih show how theprobabilities of openingand losing varyaording to the

voltage.

Therefore,theHHmodelisformedbythese fouroupledandnonlinearequations. We

emphasize thatinthis paperwe take

µ = 0

^.

3 The Integrate-and-re model

DespitetheHHmodelbeingthemostompletemodelsofar,itpresentssomedisadvantages,

as disussed previously. The Integrate-and-Fire model aims to provide spike rates in a

ertain time interval only with the information that the neuron has reahed the ation

threshold. This model has the same deterministi features as the HH model, so we also

have added whitenoise toit [3,11℄.

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The model onsists of one ODE that presents a dierent dynamis ompared to the

HH model: theneuronreahesthe predened ationthreshold

ϑ

^,^the^membrane ^potential

returnsto resting potential and theproess restarts [11℄.

τ _m dv(t)

dt = − (v(t) − E _L ) + RI(t) + R 2 , ∀ t ∈ [0, T ], T ∈ N .

⁽¹⁾

The time variation is multiplied by a time onstant

τ m

^, ^determined ^by ^the^model ^of ^one

ompartmentapaitane andbytheaverageondutanesofSodiumandleakyhannels.

We also have the onentration gradient given by the subtration of the voltage by the

resting potential

E L

^. În ^this ^model ^we âlso ^have â înput ûrrent

RI(t)

^disturbed ^by ^a

noise

R 2

^. Considering

t ^f

^to ^be ^the înstant ôf^time^that ^the^threshold îs^reahed,^we ^nish

itonsidering

v(t ^f ) = ϑ

^.

There are another two ways to perturb the IF model, but the one given by equation

(1)is theonly thatan beompared diretlyto theHH model. It ours beauseinboth

ases the noise isadded intheinputurrent.

4 Numerial Solution

The HH system and the IFdynami were disretized using nite dierene methods.

For the stohasti ases, we have used slightly dierent methods from the orresponding

deterministi ones. The omputational experiments onsist of shooting a spike train in

order to verify if the addition of noise reprodues its features more preisely. For this

purpose, HH andIFmodelswere used. All numerial experiments wereimplemented and

arriedout inMatlab

r

^R2012b ^version.

The noisy HH model was disretized using Euler-Maruyama(EM) [6℄.For values less

than

∆t = 10 ⁻ ⁴

^,^the^mean^and^the^standard^deviaton^(SD)^beomeⁱⁿ^variant. ^Furthermore, theSDonvergesto 7.92

× 10 ⁻ ⁴

^. ^By^wayôfêxample,^we ^showâdisretizationwiththeEM Method. The noise is given by dW(

t ⁿ

⁾⁼

√

∆tN (0, 1) ⁿ

^[6℄,^where ^N(0,1) ^areindependent standard Gaussian random variables.

V ⁿ ⁺¹ = V ⁿ + ∆t

I _ext − g ⁿ _{N a} − g _K ⁿ − g ⁿ _L C

+ σ 1 dW (t ⁿ );

n ⁿ ⁺¹ = n ⁿ + ∆tφ[α n (V )(1 − n) − β _n (V )n];

m ⁿ⁺¹ = m ⁿ + ∆tφ[α _m (V )(1 − m) − β _m (V )m];

h ⁿ⁺¹ = h ⁿ + ∆tφ[α _h (V )(1 − h) − β _h (V )h].

The initial onditionsand parameters aregiven by[5℄, [4℄and [1℄.

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Table1: ExperimentalparametersforHH modelwhen

µ = 0

^.

Parameter Meaning Value

C

^Membrane^spei ^apaitane ¹

µ

^F

cm ⁻ ² E _{N a}

^,

E _K

^,

E _Leak

^Nernst^potentials ^50,^-77, ^-54.4^mV

g _{N a}

^,

g _{N a}

^,

g _{N a}

^Maximum^ondutanes ^120,^36,^0.3 ^mS

cm ⁻ ²

I _ext

^Input ^urrent ¹²^mV

m

^,

n

^,

h

^Gating^variables ^0.1,^0.4, ^0.4

σ 1

^Intensity^onstant ²⁴

0 10 20 30 40 50 60 70 80 90 100

−100

−50 0 50

tempo [ms]

Voltagem [mV]

Figure1: SpiketrainwithwhitenoisethroughEMMethod forthetimeintervalof100ms.

By way of omparison to the HH model, here is the noisy IF disretization with the

EM Method[6,11℄.

V ⁿ⁺¹ = V ⁿ − ∆t

τ _m ((V ⁿ − E _L ) − RI _ext ) + σ 2 dW (t ⁿ ).

The initial onditionsand parameters aregiven by[11℄.

Table2: Experimental parametersforIFmodel.

Parameter Meaning Value

τ _m

^Time^onstant ¹⁰ ^ms

E _L

^Resting^potential ^-65^m^V

σ 2

^Intensity^onstant ²

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0 10 20 30 40 50 60 70 80 90 100

−68

−66

−64

−62

−60

−58

−56

−54

Tempo [ms]

Voltagem [mV]

Figure 2: Subthreshold dynami with white noisethroughEM Method forthe time interval of

100ms.

It an be observed that for both ases, the addition of noise has provided a more

aurate evolution oftheringneuron, byapturing its irregularities. Besides every spike

havingdierentintervalsinterspikestwobytwointherstasefortheHHandIFmodels,

the ation threshold is also diferent in a spike train (exept in the IF ase, beause the

threshold isxed). The runtime for both models alsowasompared. Using theomputer

proessorIntel(R)Core(TM i7-4790 CPU 3.60Hz)inamahinewith16384340kB of

memory,we have observed thattheIFmodelan bemore than twieasfastthan theHH

modelfor this ase.

In order to analyse the output probability distribution for the ase where

µ = 0

^, ^we

have made a Monte Carlo approah. It were realized 500 samples of spike trains, with

500 neuronringeah one,totalizing250000rings. Weexpetthat, fornon-spontaneous

rings, but with Gaussian noise input, the histogram an be well approximated by a

lognormal distribution [2,11℄.

5 10 15 20 25

0 0.5 1 1.5 2 x 10 ⁴

ISI [ms]

Numero de disparos

Figure 3: ISI-histogramforHHmodelonsidering500randomsampleswith500ringeah.

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5 10 15 20 25 30 35 40 45 50 0

5 10 15 20 25 30 35

ISI [ms]

Numero de disparos

Figure4: ISI-histogramforIFmodelonsidering500randomsampleswith500ringeah.

Table3: DispersiondataforIFmodel.

Model Timestep size Mean SD

HH

10 ⁻ ⁴

^7.744 ^2.228

IF

10 ⁻ ⁴

^11.886 ^7.659

BothISI-histogramsanbewellaproximatedbyalognormaldistribution,andaording

to the dispersion measures we observe that the HH model is loser to a exponential

distribution, whih showsthatspike trainsunderouromputationalassumption areoften

generated witha Poisson proess[11℄.

5 Conluding Remarks

The addition of standard Brownian Motion unertainty to the HH and IF models

providesmoreaurateresults. Thisanbeobserved bythefatthatthestohastimodel

reprodues inherent shape irregularities in exitatory and inhibitory ortial ells, while

the deterministi model reprodues onstant ring. One an also observe that the ISI-

histogram for the temporal asean be well approximated by alognormal distributionas

expetedfor one soureof Gaussian noise.

Referenes

[1℄ R. R. Borges, K. C. Iarosz, A. M. Batista, I. L. Caldas, F. S. Borges and E. L.

Lameu,Sinronização dedisparos emredes neuronaisomplastiidade sináptia,Re-

vista Brasileira de Ensino de Físia, 37:2310-1 - 2310-9, 2015. DOI: 10.1590/S1806-

11173721787.

[2℄ N.Bruneland X.Wang, Eetsof Neuromodulation ina CortialNetwork Modelof

Objet Working Memory Dominated by Reurrent Inhibition, Journal of Computa-

tional Neurosiene, 11:63-85, 2001.DOI: 10.1023/A:1011204814320.

(7)

[3℄ W. Gerstner, Population Dynamis of Spiking Neurons: Fast Transients, Asyn-

horonous States, and Loking, Neural Computation, 12:43-89, 2000. DOI:

10.1162/089976600300015899.

[4℄ M. Hanslien, K. H. Karlsen and A. Tveito, A maximum priniple for and expliit

nite dierene sheme approximating the Hodgkin-Huxley model, BIT Numerial

Mathematis,45:725-741, 2005.DOI: 10.1007/s10543-005-0023-2.

[5℄ G. Hermentrout and D. H. Terman, Mathematial Foundations of Neurosiene,

Springer,35:1-28, 2013.ISBN: 978-0-387-87707-5.

[6℄ D. J. Higham, An Algorithmi Introdution to Numerial Simulation of

Stohasti Dierential Equations, SIAM Review, 43:525-546, 2001. DOI:

10.1137/S0036144500378302.

[7℄ A. L. Hodgkin and A. F. Huxley, A quantitative desription of Membrane Current

and its appliation to ondution and exitation in nerve, J. Physiol., 117:500-544,

1952.DOI: 10.1113/jphysiol.1952.sp004764.

[8℄ E. M. Izhikevih, Simple model of spiking neurons, IEEE Transations on neural

networks, 14:1569-1672, 2003.DOI: 10.1109/TNN.2003.820440.

[9℄ E.M. Izhikevih,Whih Modelto UseforCortialSpikingNeurons?,IEEE Transa-

tionson neural networkns,15:1063-1070, 2004. DOI:10.1109/TNN.2004.832719.

[10℄ C. Meunier and I. Seveg, Playing devil's advoate: is this Hodgkin-Huxley

model useful?, TRENDS in Neurosiene, 25:558-563, 2002. DOI: 10.1016/S0166-

2236(02)02278-6.

[11℄ T.P.Trappenberg,Fundamentalsof ComputationalNeurosiene,OXFORDUniver-

sityPress, pages53-71, 2010. ISBN:978-0-19-956841-3.

[12℄ A.T. Valderrama, J. Witteveen, M. Navarroand J. Blom, Unertainty Propagation

inNerveImpulsesThroughtheAtionPotentialMehanism,JournalofMathematial

Neurosiene, 5:1-9,2015. DOI: 10.1186/2190-8567-5-3.

Hodgkin-Huxley model

C ∂V

∂t = µ ∂ 2 V

∂x 2 −

3

X

i=1

g i (V − E i ) + I ext + R 1 ;

I ext

R 1

i

µ

m

h

n

dm

dt = (1 − m)α m (V ) − mβ m (V );

dh

dt = (1 − h)α h (V ) − hβ h (V );

dn

dt = (1 − n)α n (V ) − nβ n (V ).

µ = 0

ϑ

τ m dv(t)

dt = − (v(t) − E L ) + RI(t) + R 2 , ∀ t ∈ [0, T ], T ∈ N .

τ m

E L

RI(t)

R 2

t f

v(t f ) = ϑ

r

∆t = 10 − 4

× 10 − 4

t n

√

∆tN (0, 1) n

V n +1 = V n + ∆t

I ext − g n N a − g K n − g n L C

+ σ 1 dW (t n );

n n +1 = n n + ∆tφ[α n (V )(1 − n) − β n (V )n];

m n+1 = m n + ∆tφ[α m (V )(1 − m) − β m (V )m];

h n+1 = h n + ∆tφ[α h (V )(1 − h) − β h (V )h].

µ = 0

C

µ

cm − 2 E N a

E K

E Leak

g N a

g N a

g N a

cm − 2

I ext

m

n

h

σ 1

0 10 20 30 40 50 60 70 80 90 100

−100

−50 0 50

tempo [ms]

Voltagem [mV]

V n+1 = V n − ∆t

τ m ((V n − E L ) − RI ext ) + σ 2 dW (t n ).

τ m

E L

σ 2

0 10 20 30 40 50 60 70 80 90 100

−68

−66

−64

−62

−60

−58

−56

−54

Tempo [ms]

Voltagem [mV]

µ = 0

∂t = µ ∂ ² V

∂x ² −

I _ext

dt = (1 − h)α h (V ) − hβ _h (V );

dt = (1 − n)α _n (V ) − nβ _n (V ).

τ _m dv(t)

dt = − (v(t) − E _L ) + RI(t) + R 2 , ∀ t ∈ [0, T ], T ∈ N .

t ^f

v(t ^f ) = ϑ

∆t = 10 ⁻ ⁴

× 10 ⁻ ⁴

t ⁿ

∆tN (0, 1) ⁿ

V ⁿ ⁺¹ = V ⁿ + ∆t

I _ext − g ⁿ _{N a} − g _K ⁿ − g ⁿ _L C

+ σ 1 dW (t ⁿ );

n ⁿ ⁺¹ = n ⁿ + ∆tφ[α n (V )(1 − n) − β _n (V )n];

m ⁿ⁺¹ = m ⁿ + ∆tφ[α _m (V )(1 − m) − β _m (V )m];

h ⁿ⁺¹ = h ⁿ + ∆tφ[α _h (V )(1 − h) − β _h (V )h].

cm ⁻ ² E _{N a}

E _K

E _Leak

g _{N a}

g _{N a}

g _{N a}

cm ⁻ ²

I _ext

V ⁿ⁺¹ = V ⁿ − ∆t

τ _m ((V ⁿ − E _L ) − RI _ext ) + σ 2 dW (t ⁿ ).

τ _m

E _L

0 0.5 1 1.5 2 x 10 ⁴

10 ⁻ ⁴

10 ⁻ ⁴