1 Introduction
Abstract

Brain plasticity refers to brain’s ability to change neuronal connections, as a result of environmental stimuli, new experiences, or damage. In this work, we study the effects of the synaptic delay on both the coupling strengths and synchronisation in a neuronal network with synaptic plasticity. We build a network of Hodgkin-Huxley neurons, where the plasticity 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 topology. Regarding the synchronisation, only for small values of the synaptic delay this phenomenon is observed.

Alterations in brain connectivity due to plasticity


and synaptic delay



E.L. Lameu, E.E.N. Macau, F.S. Borges, K.C. Iarosz, I.L. Caldas, R.R. Borges, P.R. Protachevicz, R.L. Viana, A.M. Batista
National Institute for Space Research, São José dos Campos, SP, Brazil.
Federal University of São Paulo, São José dos Campos, SP Brazil.
Physics Institute, University of São Paulo, São Paulo, SP, Brazil.
Department of Mathematics, Federal Technological University of Paraná, Apucarana, PR, Brazil.
Science Post-Graduation, State University of Ponta Grossa, Ponta Grossa, PR, Brazil.
Physics Department, Federal University of Paraná, Curitiba, PR, Brazil.
Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil.
Corresponding author: ewandson.ll@gmail.com

keywords: magnetic surfaces, sympletic map, divertor

1 Introduction

Neuroplasticity, also known as brain plasticity, refers to brain’s ability to change neuronal connections, as a result of environmental stimuli, new experiences, or damage [1]. The brain plasticity can be functional or structural. The functional plasticity occurs when functions are moved from a damaged to other undamaged areas, and structural plasticity is associated with changes in the physical structure [2]. On this regard, Borges et al. [3, 4] studied the effects of the spike timing-dependent plasticity (STDP) on the neuronal synchronisation. They observed that the transition between desynchronised and synchronised states depends on the external perturbation level and the neuronal architecture. It is know that neuronal synchronisation is important in information binding [5] and cognitive functions [6]. Nevertheless, synchronisation can be related to brain disorders such as Parkinson’s disease [7] and seizures [8]. This way, there have been many researches about not only neuronal synchronisation [9], but also suppression of synchronous behaviour [10].

We focus here on the effects of the synaptic delay on a neuronal network with STDP. Information transmission delay is inherent due to both the delays in synaptic transmission and the finite propagation velocities in the conduction of signals [11]. Hao et al. [12] studied synchronisation transitions in a modified Hodgkin-Huxley neuronal network with time delay. They found multiple synchronisation 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 of experiments on rhesus monkeys. More significant evidence began to be observed in the 1960s. In 1964, Diamond et al. [14, 15] published research about neuroplasticity, which is considered as the first evidence of anatomical brain plasticity. Bach-y-Rita [16] created a machine that helped blind people not only to distinguish objects, but also to read. In 1949, the neuropsychologist Donald Olding Hebb [17] wrote a book entitled “The organization of behavior”, where he proposed that neurons which fire together, also wire together. The Hebbian plasticity led model of spike timing-dependent plasticity (STDP). The STDP function for excitatory and inhibitory synapses were showed by Bi and Poo [18] and Haas et al. [19], respectively.

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

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

2 Hodgkin-Huxley neural network with synaptic delay

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

(1)
(2)
(3)
(4)

where (F/cm) is the membrane capacitance and (mV) is the membrane potential of neuron (). represents a constant current density that is randomly distributed in the interval , (excitatory) and (inhibitory) are the average degree connectivities, and are the excitatory and inhibitory coupling strengths from the presynaptic neuron to the postsynaptic neuron . and are the number of excitatory and inhibitory neurons, respectively. The parameters and are the condutances of the potassium, sodium and leak ion channels, respectively. and are the reversal potentials for these ion channels. The functions and represent the activation for sodium and potassium, respectively. is the function for the inactivation of sodium. The functions , , , ,, are given by

(5)
(6)
(7)
(8)
(9)
(10)

where . The neuron can present periodic spikings or single spike activity as a result of the variation of the external current density (A/cm). The frequency of the periodic spikes increases if the constant increases.

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

(11)

where is the synaptic time constant and is the most recent firing instant of the neuron . The parameter is the time delay and consequently the time that the current spends to achieve the postsynaptic neuron [12]. Figures 1(a) and 1(c) show the time evolution of the action potential for and ms, respectively. The action potential starts at mV and when a stimulus is applied it spikes upward. After the peak potential, the action potential falls to the resting potential. In Figures 1(b) and 1(d) we calculate for the respective Figures 1(a) and 1(c). We see by means of the dashed green line that the transmission of the synaptic current to the postsynaptic is not instantaneous for ms.

In our simulations, we consider F/cm, mV, mV, mV, mS/cm, mS/cm, mS/cm and ms. The neurons are excitatorily coupled with a reversal potential mV, and inhibitorily coupled with a reversal potential mV [4].

Figure 1: Time evolution of the action potential of a presynaptic neuron and the respective synaptic current (output) that achieves the postsynaptic neuron . We consider in (a) and (b), and 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, the neuroscientists Bi and Poo [18] characterised the dependence of the long-term potentiation and depression on the order and timing of pre and postsynaptic spikes, named spike time dependant plasticity (STDP). The plasticity dynamics is given by the update value of the synaptic weight , and a mathematical definition of this function is given by [21]

(12)

Kalitzin and collaborators [21] showed that the function depends on the membrane potential of the postsynaptic neuron, the activation of the synapse, and the thresholds for switching on long-term potentiation and the long-term depression. We consider an approximation of in the linear form [4]. The function is the solution of Equation (12), where , , and are constants. For , we obtain the update value for excitatory synapses (eSTDP), and for , we find the update value for inhibitory synapses (iSTDP). The plasticity dynamics introduced by means of this linear approximation is not related to physiological processes [22], however, with this function we can find a fit which describes experimental results of eSTDP and iSTDP, as showed in References [18] and [19].

Figure 2(a) exhibits the eSTDP function for excitatory synapses, where the presynaptic neuron and the postsynaptic neuron are forced to spike at time and , respectively. There is a change in the synaptic weights due to the time difference between the spikes . The eSTDP function is given by [23]

(13)

where , , ms, and ms. The synaptic weights are updated according to Equation (13), where . The black line in figure 2(a) shows the potentiation of excitatory synaptic weights for and the blue line the depression in synaptic weights for .

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. The weights are increased based on the following equation

(14)

where , , if , if and [24, 25]. The inhibitory synaptic weights are updated according to Equation (14), where .

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

4 Synaptic weights and synchronisation

In our simulations, aiming to understand the alterations in network connectivity, we consider a neuronal network with Hodgkin-Huxley. This number of neurons was chosen to facilitate a visual analysis of the coupling matrices without to lose dynamics properties. Our network has of excitatory and of inhibitory synapses according to anatomical estimates for the neocortex [26]. The neurons are initially globally coupled and the initial synaptic weights are normally distributed with mean and standard deviation equal to . 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 see the coupling matrices, where the colour bar represents the synaptic weights. The coupling matrix is separated into excitatory () and inhibitory () neurons. The excitatory neurons are organised from the lowest frequency to the highest frequency , and the inhibitory neurons from the lowest frequency to the highest frequency .

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

Figure 3(a) exhibits the initial synaptic weights separated into regions. In the regions I and II the synapses from the pre to the postsynaptic neurons are excitatory. The region III and IV have inhibitory synapses from the pre to postsynaptic neurons. For ms, we observe in Figure 3(b) that the coupling matrix shows a triangular shape, due to the fact that the excitatory synapses become stronger from the high to low frequency neurons and the inhibitory synapses from the low to high frequency neurons. When the time delay is ms and also ms, as shown in Figures 3(c) and 3(d), respectively, the coupling matrices have a non-trivial configuration of connections, presenting a greater agreement with real neuronal networks [27, 28, 29]. Therefore, the time delay has a significant influence on the synaptic weights in a neuronal network with plasticity, resulting in non-trivial configurations and synaptic weights with greater variability in their values if compared to the case without delay

We analyse the time evolution of instantaneous average of excitatory and inhibitory coupling strengths for different time delay values. Without time delay (Figure 4(a)), (black line) has value greater than (red line). Whereas for ms (Figure 4(b)) and ms (Figure 4(c)) both and oscillate in the interval .

Figure 4: In this set of Figures we show the time evolution of (black line) and (red line) for (a) , (b) ms, and (c) ms, as well as the time evolution of the Kuramoto’s order parameter for(d) , (e) ms, and (f) ms.

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

(15)

and the time averaged order parameter

(16)

where is the phase associated with the spikes,

(17)

where is the time windows for measuring, is the time when a spike () in the neuron happens (). The order parameter magnitude asymptotes to unity when the network has a globally synchronised behaviour. For uncorrelated spiking phases, the order parameter is much less than 1.

Figures 4(d), 4(e), and 4(f) exhibit the order parameter for (d) , (e) ms, and (f) ms. Our neuronal network does not exhibit completely synchronisation due to the fact that the neurons are not identical. Nevertheless, for the neuronal network shows strong synchronisation behaviour. In Figure 4(d), we see a synchronous state for . There is no synchronisation states observed for ms and ms, as shown in Figures 4(e) and 4(f), respectively. This result shows that the delay is an important mechanism in the network dynamics, avoiding synchronization.

In Figures 5(a) we calculate the time averaged excitatory and inhibitory coupling strengths as a function of the time delay for different initial conditions. The values presents a small variation as the delay is increased. However, is more sensitive and for small delay values ms we observe and the network is more excitable. As a result the neurons in the network are strongly synchronized (Figure 5(b)). When we increase the delay for ms the values of starts to decrease in a second order transition. Simultaneously the order parameter decreases showing its dependence with the excitatory coupling strength . Finally, for ms we observe that and oscillates in the interval and the network are no longer synchronized. These results show us that synchronization in a neuronal network with plasticity and synaptic delay is closely linked to the intensity of excitatory couplings, i.e, the more excitable the network () the more synchronous the neurons will be.

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

5 Conclusion

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

We carry out simulations considering a coupling matrix with initial synaptic weights normally distributed. Without time delay, the coupling matrix evolves to a triangular shape, where the excitatory synapses are stronger from the high frequency to low frequency neurons an the inhibitory synapses increases in the opposite way. The coupling matrix exhibits non-trivial configuration when the time delay is increased.

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

Acknowledgments

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

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