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Alterations in brain connectivity due to plasticityand synaptic delay

E.L. Lameu1, E.E.N. Macau1,2, F.S. Borges3, K.C. Iarosz3, I.L. Caldas3, R.R.Borges4, P.R. Protachevicz5, R.L. Viana6, A.M. Batista3,5,71National Institute for Space Research, Sao Jose dos Campos, SP, Brazil.2Federal University of Sao Paulo, Sao Jose dos Campos, SP Brazil.3Physics Institute, University of Sao Paulo, Sao Paulo, SP, Brazil.4Department of Mathematics, Federal Technological University of Parana, Apu-carana, PR, Brazil.5Science Post-Graduation, State University of Ponta Grossa, Ponta Grossa, PR,Brazil.6Physics Department, Federal University of Parana, Curitiba, PR, Brazil.7Department of Mathematics and Statistics, State University of Ponta Grossa,Ponta Grossa, PR, Brazil.Corresponding author: ewandson.ll@gmail.com

Abstract

Brain plasticity refers to brain’s ability to change neuronal connec-

tions, as a result of environmental stimuli, new experiences, or damage.

In this work, we study the effects of the synaptic delay on both the cou-

pling strengths and synchronisation in a neuronal network with synaptic

plasticity. We build a network of Hodgkin-Huxley neurons, where the plas-

ticity is given by the Hebbian rules. We verify that without time delay

the excitatory synapses became stronger from the high frequency to low

frequency neurons and the inhibitory synapses increases in the opposite

way, when the delay is increased the network presents a non-trivial topol-

ogy. Regarding the synchronisation, only for small values of the synaptic

delay this phenomenon is observed.

keywords: magnetic surfaces, sympletic map, divertor

1 Introduction

Neuroplasticity, also known as brain plasticity, refers to brain’s ability to changeneuronal connections, as a result of environmental stimuli, new experiences, ordamage [1]. The brain plasticity can be functional or structural. The functionalplasticity occurs when functions are moved from a damaged to other undam-aged areas, and structural plasticity is associated with changes in the physicalstructure [2]. On this regard, Borges et al. [3, 4] studied the effects of the spiketiming-dependent plasticity (STDP) on the neuronal synchronisation. Theyobserved that the transition between desynchronised and synchronised statesdepends on the external perturbation level and the neuronal architecture. It isknow that neuronal synchronisation is important in information binding [5] andcognitive functions [6]. Nevertheless, synchronisation can be related to braindisorders such as Parkinson’s disease [7] and seizures [8]. This way, there have

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been many researches about not only neuronal synchronisation [9], but alsosuppression of synchronous behaviour [10].

We focus here on the effects of the synaptic delay on a neuronal networkwith STDP. Information transmission delay is inherent due to both the delaysin synaptic transmission and the finite propagation velocities in the conductionof signals [11]. Hao et al. [12] studied synchronisation transitions in a modi-fied Hodgkin-Huxley neuronal network with time delay. They found multiplesynchronisation transitions when the time delay is considered.

Experimental evidence of neuroplasticity was provide by Lashely in 1923[13]. He dentified high evidence of changes in neural pathways by means ofexperiments on rhesus monkeys. More significant evidence began to be ob-served in the 1960s. In 1964, Diamond et al. [14, 15] published research aboutneuroplasticity, which is considered as the first evidence of anatomical brainplasticity. Bach-y-Rita [16] created a machine that helped blind people notonly to distinguish objects, but also to read. In 1949, the neuropsychologistDonald Olding Hebb [17] wrote a book entitled “The organization of behavior”,where he proposed that neurons which fire together, also wire together. TheHebbian plasticity led model of spike timing-dependent plasticity (STDP). TheSTDP function for excitatory and inhibitory synapses were showed by Bi andPoo [18] and Haas et al. [19], respectively.

In this work, our results suggest that alterations in the synchronisation andconnectivity in a plastic network depend on the synaptic delay. We consider aHodgkin-Huxley neuronal network with inhibitory and excitatory neurons. TheHodgkin-Huxley model [20] was proposed in 1952, and it is given by coupleddifferential equations that explains the ionic mechanisms.

This paper is organised as follows: Section 2 introduces the Hodgkin-Huxleyneural network with synaptic delay. In Section 3, we introduce the synapticplasticity. In Section 4, we show our results about synaptic weights and neuronalsynchronisation. In the last Section, we draw the conclusions.

2 Hodgkin-Huxley neural network with synap-

tic delay

In the neuronal network we consider as local dynamics the neuron model pro-posed by Hodgkin and Huxley in 1952 [20]. The individual dynamics of eachneuron in the network is given by

CVi = Ii − gKn4i (Vi − EK)− gNam

3ihi(Vi − ENa)− gL(Vi − EL)

+(V Exc

r − Vi)

ωExc

NExc∑

j=1

εijfj(t) +(V Inhib

r − Vi)

ωInhib

NInhib∑

j=1

σijfj(t), (1)

ni = αni(Vi)(1 − ni)− βni

(Vi)ni, (2)

mi = αmi(Vi)(1−mi)− βmi

(Vi)mi, (3)

hi = αhi(Vi)(1− hi)− βhi

(Vi)hi, (4)

2

where C (µF/cm2) is the membrane capacitance and Vi (mV) is the membranepotential of neuron i (i = 1, ..., N). Ii represents a constant current density thatis randomly distributed in the interval [9.0; 10.0], ωExc (excitatory) and ωInhib

(inhibitory) are the average degree connectivities, εij and σij are the excitatoryand inhibitory coupling strengths from the presynaptic neuron j to the postsy-naptic neuron i. NExc and NInhib are the number of excitatory and inhibitoryneurons, respectively. The parameters gK , gNa and gL are the condutances ofthe potassium, sodium and leak ion channels, respectively. EK , ENa and EL arethe reversal potentials for these ion channels. The functions m(Vi) and n(Vi)represent the activation for sodium and potassium, respectively. h(Vi) is thefunction for the inactivation of sodium. The functions αn, βn, αm, βm,αh, βn

are given by

αn(v) =0.01v + 0.55

1− exp (−0.1v − 5.5), (5)

βn(v) = 0.125 exp

(

−v − 65

80

)

, (6)

αm(v) =0.1v + 4

1− exp (−0.1v − 4), (7)

βm(v) = 4 exp

(

−v − 65

18

)

, (8)

αh(v) = 0.07 exp

(

−v − 65

20

)

, (9)

βh(v) =1

1 + exp (−0.1v − 3.5), (10)

where v = V/[mV ]. The neuron can present periodic spikings or single spikeactivity as a result of the variation of the external current density Ii (µA/cm

2).The frequency of the periodic spikes increases if the constant Ii increases.

In Equation (1) the term fj(t) is a function which represents the strengthof an effective synaptic (output) current and it is given by

fj(t) = e−(t−tj−τ)

τs , (11)

where τs is the synaptic time constant and tj is the most recent firing instantof the neuron j. The parameter τ is the time delay and consequently the timethat the current fj(t) spends to achieve the postsynaptic neuron [12]. Figures1(a) and 1(c) show the time evolution of the action potential Vj(t) for τ = 0and τ = 3 ms, respectively. The action potential starts at −70 mV and whena stimulus is applied it spikes upward. After the peak potential, the actionpotential falls to the resting potential. In Figures 1(b) and 1(d) we calculatefj(t) for the respective Figures 1(a) and 1(c). We see by means of the dashedgreen line that the transmission of the synaptic current to the postsynaptic isnot instantaneous for τ = 3 ms.

In our simulations, we consider C = 1 µ F/cm2, ENa = 50 mV, EK =−77 mV, EL = −54.4 mV, gNa = 120 mS/cm2, gK = 36 mS/cm2, gL = 0.3

3

mS/cm2 and τs = 2.728 ms. The neurons are excitatorily coupled with a reversalpotential V Exc

r = 20 mV, and inhibitorily coupled with a reversal potentialV Inhibr = −75 mV [4].

-80

-40

0

40

Vj(t)

400 425 450t (ms)

0

0.5

1

fj(t)

400 425 450t (ms)

(a)

(b)

(c)

(d)

Figure 1: Time evolution of the action potential Vj(t) of a presynaptic neuron jand the respective synaptic current (output) fj(t) that achieves the postsynapticneuron i. We consider τ = 0.0 in (a) and (b), and τ = 3.0 ms in (c) and (d).

3 Synaptic plasticity

Synaptic plasticity is the process that produces changes in the synaptic strength,namely it is the strengthening or weakening of synapses over time. In 1998, theneuroscientists Bi and Poo [18] characterised the dependence of the long-termpotentiation and depression on the order and timing of pre and postsynapticspikes, named spike time dependant plasticity (STDP). The plasticity dynamicsis given by the update value of the synaptic weight ∆Γ, and a mathematicaldefinition of this function is given by [21]

d∆Γ(t)

dt= y(∆Γ, V, t). (12)

Kalitzin and collaborators [21] showed that the function y depends on the mem-brane potential of the postsynaptic neuron, the activation of the synapse, andthe thresholds for switching on long-term potentiation and the long-term de-pression. We consider an approximation of y in the linear form y(∆Γ, t) =(a+ c/t)∆Γ [4]. The function ∆Γ = btc exp(at) is the solution of Equation (12),where a, b, and c are constants. For c = 0, we obtain the update value forexcitatory synapses ∆ε (eSTDP), and for c 6= 0, we find the update value forinhibitory synapses ∆σ (iSTDP). The plasticity dynamics introduced by means

4

of this linear approximation is not related to physiological processes [22], how-ever, with this function we can find a fit which describes experimental resultsof eSTDP and iSTDP, as showed in References [18] and [19].

Figure 2(a) exhibits the eSTDP function for excitatory synapses, where thepresynaptic neuron j and the postsynaptic neuron i are forced to spike at timetj and ti, respectively. There is a change in the synaptic weights ∆εij due tothe time difference between the spikes ∆tij = ti − tj . The eSTDP function isgiven by [23]

∆εij =

{

A1 exp(−∆tij/τ1), if ∆tij ≥ 0−A2 exp(∆tij/τ2), if ∆tij < 0

, (13)

where A1 = 1, A2 = 0.5, τ1 = 1.8ms, and τ2 = 6ms. The synaptic weights areupdated according to Equation (13), where εij → εij+10−3∆εij . The black linein figure 2(a) shows the potentiation of excitatory synaptic weights for ∆tij ≥ 0and the blue line the depression in synaptic weights for ∆tij < 0.

0

0.5

1

|∆εij|

potentiationdepression

0 5 10 15 20|∆t

ij| (ms)

0

0.5

1

|∆σij|

(a)

(b)

Figure 2: Comparison between absolute values of potentiation (black curves)versus depression (blue curves) in synaptic weights. STDP function for (a)excitatory and (b) inhibitory synapses.

In Figure 2(b), we see the iSTDP function for inhibitory synapses. Theweights are increased based on the following equation

∆σij =g0

gnormαβ |∆tij |∆tij

β−1 exp(−α|∆tij |), (14)

where g0 = 0.02, β = 10, α = 0.94 if ∆tij > 0, α = 1.1 if ∆tij < 0 andgnorm = ββ exp(−β) [24, 25]. The inhibitory synaptic weights are updatedaccording to Equation (14), where σij → σij + 10−3∆σij .

5

In our neural network model, the time interval between spikes ∆tij and theplasticity rules are calculated and applied every time the postsynaptic neuron ifires and can present different values depending on when the presynaptic neuronj had the last spike.

4 Synaptic weights and synchronisation

In our simulations, aiming to understand the alterations in network connectiv-ity, we consider a neuronal network with 100 Hodgkin-Huxley. This numberof neurons was chosen to facilitate a visual analysis of the coupling matriceswithout to lose dynamics properties. Our network has 80% of excitatory and20% of inhibitory synapses according to anatomical estimates for the neocor-tex [26]. The neurons are initially globally coupled and the initial synapticweights are normally distributed with mean 0.25 and standard deviation equalto 0.02. In this approach, to understand the impact of the delay in the system,we will consider that all the synapses have the same delay. In Figure 3 we seethe coupling matrices, where the colour bar represents the synaptic weights.The coupling matrix is separated into excitatory (1 ≤ i, j ≤ 80) and inhibitory(81 ≤ i, j ≤ 100) neurons. The excitatory neurons i are organised from the low-est frequency i = 1 to the highest frequency i = 80, and the inhibitory neuronsfrom the lowest frequency i = 81 to the highest frequency i = 100.

Figure 3(a) exhibits the initial synaptic weights separated into 4 regions.In the regions I and II the synapses from the pre to the postsynaptic neuronsare excitatory. The region III and IV have inhibitory synapses from the pre topostsynaptic neurons. For τ = 0ms, we observe in Figure 3(b) that the couplingmatrix shows a triangular shape, due to the fact that the excitatory synapsesbecome stronger from the high to low frequency neurons and the inhibitorysynapses from the low to high frequency neurons. When the time delay isτ = 3ms and also τ = 6ms, as shown in Figures 3(c) and 3(d), respectively, thecoupling matrices have a non-trivial configuration of connections, presenting agreater agreement with real neuronal networks [27, 28, 29]. Therefore, the timedelay has a significant influence on the synaptic weights in a neuronal networkwith plasticity, resulting in non-trivial configurations and synaptic weights withgreater variability in their values if compared to the case without delay

We analyse the time evolution of instantaneous average of excitatory ε(t)and σ(t) inhibitory coupling strengths for different time delay values. Withouttime delay τ = 0 (Figure 4(a)), ε (black line) has value greater than σ (red line).Whereas for τ = 3ms (Figure 4(b)) and τ = 6ms (Figure 4(c)) both ε and σoscillate in the interval [0.2; 0.3].

We study the effects of the time delay on the neuronal synchronisation. Todo that, we use the Kuramoto order parameter as diagnostic tool, that is givenby [30]

R(t) =

∣

∣

∣

∣

∣

∣

1

N

N∑

j=1

exp(iφj(t))

∣

∣

∣

∣

∣

∣

, (15)

6

1

80

100

1 80 100

Ne

uro

n (

j)

Neuron (i)

0

0.25

0.5

Syn

ap

tic W

eig

hts

1

80

100

1 80 100

Neuro

n (

j)

Neuron (i)

0

0.25

0.5

Synaptic W

eig

hts

1

80

100

1 80 100

Ne

uro

n (

j)

Neuron (i)

0

0.25

0.5

Syn

ap

tic W

eig

hts

1

80

100

1 80 100

Neuro

n (

j)

Neuron (i)

0

0.25

0.5

Synaptic W

eig

hts

(a) (b)

(c) (d)

I

III IV

II

Figure 3: Coupling matrices for excitatory and inhibitory neurons. Figure (a)shows the initial synaptic weights. We consider (b) τ = 0 ms, (c) τ = 3 ms and(d) τ = 6 ms at 400 s. In four cases the colour bar represents the synapticsweights.

7

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4

t (105ms)

0

0.2

0.4

0.6

0.8

1

R(t)

1 2 3 4

t (105ms)

1 2 3 4

t (105ms)

ε(t)σ(t)

(a) (b) (c)

(d) (e) (f)

Figure 4: In this set of Figures we show the time evolution of ε(t) (black line)and σ(t) (red line) for (a) τ = 0, (b) τ = 3 ms, and (c) τ = 6 ms, as well as thetime evolution of the Kuramoto’s order parameter R(t) for(d) τ = 0, (e) τ = 3ms, and (f) τ = 6 ms.

8

and the time averaged order parameter

R =1

tf − ti

tf∑

ti

∣

∣

∣

∣

∣

∣

1

N

N∑

j=1

exp(iφj(t))

∣

∣

∣

∣

∣

∣

, (16)

where φj(t) is the phase associated with the spikes,

φj(t) = 2πm+ 2πt− tj,m

tj,m+1 − tj,m, (17)

where tf − ti is the time windows for measuring, tj,m is the time when a spikem (m = 0, 1, 2, . . .) in the neuron j happens (tj,m < t < tj,m+1). The orderparameter magnitude asymptotes to unity when the network has a globallysynchronised behaviour. For uncorrelated spiking phases, the order parameteris much less than 1.

Figures 4(d), 4(e), and 4(f) exhibit the order parameter for (d) τ = 0, (e)τ = 3ms, and (f) τ = 6ms. Our neuronal network does not exhibit completelysynchronisation due to the fact that the neurons are not identical. Nevertheless,for R > 0.9 the neuronal network shows strong synchronisation behaviour. InFigure 4(d), we see a synchronous state for τ = 0. There is no synchronisationstates observed for τ = 3ms and τ = 6ms, as shown in Figures 4(e) and 4(f),respectively. This result shows that the delay is an important mechanism in thenetwork dynamics, avoiding synchronization.

In Figures 5(a) we calculate the time averaged excitatory and inhibitory cou-pling strengths as a function of the time delay for 10 different initial conditions.The σ values presents a small variation as the delay τ is increased. However, εis more sensitive and for small delay values τ < 1.5ms we observe ε > σ and thenetwork is more excitable. As a result the neurons in the network are stronglysynchronized (Figure 5(b)). When we increase the delay for τ > 1.5 ms thevalues of ε starts to decrease in a second order transition. Simultaneously theorder parameter R decreases showing its dependence with the excitatory cou-pling strength ε. Finally, for τ > 2.5 ms we observe that ε and σ oscillates inthe interval [0.2; 0.3] and the network are no longer synchronized. These resultsshow us that synchronization in a neuronal network with plasticity and synap-tic delay is closely linked to the intensity of excitatory couplings, i.e, the moreexcitable the network (ε > σ) the more synchronous the neurons will be.

5 Conclusion

We study a neural network with plasticity and synaptic delay, where we considerthe Hodgkin-Huxley model as local dynamics. The Hodgkin-Huxley neuron isa mathematical model described by coupled differential equations that exhibitsspiking dynamics. We build a network with an initial all-to-all topology andanalyse the time evolution of the connectivity and synchronisation.

We carry out simulations considering a coupling matrix with initial synapticweights normally distributed. Without time delay, the coupling matrix evolves

9

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6τ0

0.2

0.4

0.6

0.8

1

R

εσ

(a)

(b)

Figure 5: (a) Average values of excitatory ε, inhibitory σ synaptic weights, and(b) mean order parameter R as a function of synaptic time delay τ . The barsshow the standard deviation from the mean values.

10

to a triangular shape, where the excitatory synapses are stronger from the highfrequency to low frequency neurons an the inhibitory synapses increases in theopposite way. The coupling matrix exhibits non-trivial configuration when thetime delay is increased.

We also show that the time delay plays an important role in the neuralsynchronisation. Increasing the time delay, we verify that the time averagedexcitatory coupling strength decrease and it becomes approximately equal tothe averaged inhibitory coupling strength. As a consequence, this decreasesuppresses the synchronous behaviour of the neural network.

Acknowledgments

This work was possible by partial financial support from the following Braziliangovernment agencies: CNPq (154705/2016-0, 311467/2014-8), CAPES, Funda-cao Araucaria, and Sao Paulo Research Foundation (processes FAPESP 2011/19296-1, 2015/ 07311-7, 2016/23398-8, 2015/50122-0). Research supported by grant2015/50122-0Sao Paulo Research Foundation (FAPESP) and DFG-IRTG 1740/2.

References

[1] A. Pascual-Leone, A. Amedi, F. Fregni, L.B. Merabet, Annu. Rev. Neu-rosci. 28, 377 (2005)

[2] B. Kolb, R. Gibb, J. Can. Acad. Child Adolesc. Psychiatry 20, 265 (2011)

[3] R.R. Borges, F.S. Borges, E.L. Lameu, A.M. Batista, K.C. Iarosz, I.L. Cal-das, R.L. Viana, M.A.F. Sanjuan, Commun. Nonlinear Sci. Numer. Simul.34, 12 (2016)

[4] R.R. Borges, F.S. Borges, E.L. Lameu, A.M. Batista, K.C. Iarosz, I.L.Caldas, C.G. Antonopoulos, M.S. Baptista. Neural Netw. 88, 58 (2017)

[5] R. Lestienne, Prog. Neurobiol. 65, 545 (2001)

[6] X.-J. Wang, Physiol. Rev. 90, 1195 (2010)

[7] B.C. Schwab, T. Heida, Y. Zhao, E. Marani, S.A. van Gils, R.J.A. vanWezel, Front. Syst. Neurosci. 7, 60 (2013)

[8] S. Boucetta, S. Chauvette, M. Bazhenov, I. Timofeev, Epilepsia 49, 1925(2008)

[9] F.S. Borges, P.R. Protachevicz, E.L. Lameu, R.C. Bonetti, K.C. Iarosz,I.L. Caldas, M.S. Baptista, A.M. Batista, Neural Netw. 90, 1 (2017)

[10] E.L. Lameu, F.S. Borges, R.R. Borges, K.C. Iarosz, I.L. Caldas, A.M.Batista, R.L. Viana, J. Kurths, Chaos 26, 043107 (2016)

11

[11] E.R. Kandel, J.H. Schwartz, T.M. Jessel, Principles of neural science (El-sevier, Amsterdam, 1991)

[12] Y. Hao, Y. Gong, L. Wang, X. Ma, C. Yang, Chaos Sol. Fractals 44, 260(2011)

[13] K. Lashley, Psychol. Bull. 30, 237 (1923)

[14] M.C. Diamond, D. Krech, M.R. Rosenzweig, J. Comp. Neurol. 123, 111(1964)

[15] E.L. Bennett, M.C. Diamond, D. Krech, M.R. Rosenzweig, Science 146,610 (1964)

[16] P. Bach-y-Rita, Acta Neurol Scandinav. 43, 417 (1967)

[17] D.O. Hebb, The organization of behavior (Wiley, New York, 1949)

[18] G.Q. Bi, M.M. Poo, J. Neurosci. 18, 10464 (1998)

[19] J.S. Haas, T. Nowotny, H.D.I. Abarbanel, J. Neurophysiol. 96, 3305 (2006)

[20] A.L. Hodgkin, A.F. Huxley, J. Physiol. 11, 500 (1952)

[21] S. Kalitzin, B. W. Van Dijk, H. Spekreijse, Biol. Cybernetics 83, 139 (2000)

[22] A. Artola, S. Brocher, W. Singer, Nature 347, 69 (1990)

[23] G.Q. Bi, M.M. Poo, Annu. Rev. Neurosci. 24, 139 (2001)

[24] S.S. Talathi, D.U. Hwang, W.L. Ditto, J. Comput. Neurosci. 25, 262 (2008)

[25] H.D.I. Abarbanel, S.S. Talathi, Phys. Rev. Lett. 96, 148104 (2006)

[26] C. R. Noback, N. L. Strominger, R. J. Demarest, D. A. Ruggiero,The Hu-man Nervous System: Structure and Function (Totowa, NJ: Humana Press,(2005)

[27] S. Rieubland, A. Roth, M. Hausser, Neuron 4, 913 (2014)

[28] S. Yu, D. Huang, W. Singer, D. Nikolic, Cereb. Cortex 18, 2891 (2008)

[29] P. Bonifazi, M. Goldin, M. A. Picardo, I. Jorquera, A. Cattani, G. Bianconi,A. Represa, Y. Ben-Ari, R. Cossart, Science 326, 1419 (2009)

[30] Y. Kuramoto, Chemical oscillations, waves and turbulence (Spring-Verlag,Berlin, 1984).

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