Influence of autapses on synchronisation in neural networks with chemical synapses
P. R. Protachevicz, K. C. Iarosz, I. L. Caldas, C. G. Antonopoulos, A. M. Batista, J. Kurths
11 Influence of autapses on synchronisation in neural networks with chemical synapses
P. R. Protachevicz , K. C. Iarosz , , ∗ , I. L. Caldas , C. G. Antonopoulos , A. M. Batista , , J. Kurths , , Institute of Physics, University of S˜ao Paulo, S˜ao Paulo, SP, Brazil Faculdade de Telˆemaco Borba, FATEB, Telˆemaco Borba, Paran´a, Brazil Graduate Program in Chemical Engineering, Federal University of Technology Paran´a, Ponta Grossa, Paran´a, Brazil Department of Mathematical Sciences, University of Essex, Wivenhoe Park, UK Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, PR, Brazil Department Complexity Science, Potsdam Institute for Climate Impact Research, Potsdam, Germany Department of Physics, Humboldt University, Berlin, Germany Centre for Analysis of Complex Systems, Sechenov First Moscow State Medical University, Moscow, RussiaCorresponding author: [email protected].
Abstract
A great deal of research has been devoted on the investigation of neural dynamics in various network topologies. However,only a few studies have focused on the influence of autapses, synapses from a neuron onto itself via closed loops, on neuralsynchronisation. Here, we build a random network with adaptive exponential integrate-and-fire neurons coupled with chemicalsynapses, equipped with autapses, to study the effect of the latter on synchronous behaviour. We consider time delay in theconductance of the pre-synaptic neuron for excitatory and inhibitory connections. Interestingly, in neural networks consisting ofboth excitatory and inhibitory neurons, we uncover that synchronous behaviour depends on their synapse type. Our results provideevidence on the synchronous and desynchronous activities that emerge in random neural networks with chemical, inhibitory andexcitatory synapses where neurons are equipped with autapses.
Keywords: synapses, autapses, excitatory and inhibitory neural networks, integrate-and-fire model, neural dynamics, synchronisation
I. I
NTRODUCTION
An important research subject in neuroscience is to understand how cortical networks avoid or reach states of highsynchronisation [24]. In normal activity, excitatory and inhibitory currents are well balanced [1], [2], while in epileptic seizures,high synchronous behaviour has been related to unbalanced current inputs [3], [4]. Nazemi et al. [5] showed that the structuralcoupling strength is important for the appearance of synchronised activities in excitatory and inhibitory neural populations.Various studies discuss the relation between structure and function in microscale and macroscale brain networks [6], [7],[8], [9]. In a microscale organisation, local excitatory and inhibitory connections are responsible for a wide range of neuralinteractions [10], [11]. Bittner et al. [12] investigated population activity structure as a function of neuron types. They verifiedthat the population activity structure depends on the ratio of excitatory to inhibitory neurons sampled. The pyramidal cell(excitatory neuron) exhibit spike adaptation, while the fast spiking cell (inhibitory neuron) have a small or inexistent spikeadaptation [13], [14].The excitatory to inhibitory and inhibitory to excitatory connections can change firing rates, persistent activities andsynchronisation of the population of postsynaptic neurons [15], [16], [17], [18], [19]. Deco et al. [20] analysed the effectof control in the inhibitory to excitatory coupling on the neural firing rate. Mejias et al. [21] proposed a computational modelfor the primary cortex in which different layers of excitatory and inhibitory connections were considered.A number of studies reported that excitatory synapses facilitate neural synchronisation [22], [23], while inhibitory synapseshave an opposite effect [24], [25], [26]. The time delay related to excitatory and inhibitory synapses influences the neuralsynchronisation [27], [28]. Further on, there is a strong research interest in the investigation of how excitatory and inhibitorysynapses influence synchronisation in neural networks [29]. On the other hand, different types of networks have been used toanalyse neural synchronisation, such as random [30], [31], small-world [32], [33], [34], [35], [36], [37], regular [38], [39], andscale-free [40], [41].Experiments showed that autapses are common in the brain and that they play an important role in neural activity [42], [43],[44]. An autapse is a synaptic contact from a neuron to itself via a closed loop [45], [46], i.e. an auto-connection with a timedelay on signal transmission [48]. Although, autaptic connections are anatomically present in vivo and in the neocortex, theirfunctions are not completely understood [49]. Experimental and theoretical studies on excitatory and inhibitory autapses havebeen carried out [50], [51], [52], [53] and the results have demonstrated that autaptic connections play a significant role innormal and abnormal brain dynamics [54], [55], [56], [57]. The effects of autapses on neural dynamics were studied for singleneurons [58], [59], [60], [61] and for neural networks [62]. It has been shown that excitatory autapses contribute to a positivefeedback [63] and can maintain persistent activities in neurons [45]. It was also found that they promote burst firing patterns[64], [65]. The inhibitory autapses contribute to a negative feedback [49], [63] and to the reduction of neural excitability [66],[67], [53]. Guo et al. [69] analysed chemical and electrical autapses in the regulation of irregular neural firing. In this way,autaptic currents can modulate neural firing rates [49]. Wang et al. [70] demonstrated that chemical autapses can induce afiltering mechanism in random synaptic inputs. Interestingly, inhibitory autapses can favour synchronisation during cognitiveactivities [71]. Short-term memory storage was observed by Seung et al. [72] in a neuron with autapses submitted to excitatory a r X i v : . [ q - b i o . N C ] N ov and inhibitory currents. Finally, a study on epilepsy has exhibited that the number of autaptic connections can be different inher epileptic tissue [49].Here, we construct a random network with adaptive exponential integrate-and-fire (AEIF) neurons coupled with chemicalsynapses. The model of AEIF neurons was proposed by Brette and Gerstner [73] and has been used to mimic neural spikeand burst activities. Due to the fact that the chemical synapses can be excitatory and inhibitory, we build a network withexcitatory synapses and autapses, a network with inhibitory synapses and autapses, and a network with both types of synapsesand autapses. In the mixed network, we consider 80% of excitatory and 20% of inhibitory synapses and autapses. In thiswork, we focus on the investigation of the influence of autapses on neural synchronisation. Ladenbauer et al. [74] studiedthe role of adaptation in excitatory and inhibitory populations of AEIF neurons upon synchronisation, depending on whetherthe recurrent synaptic excitatory or inhibitory couplings dominate. In our work, we show that not only the adaptation, butalso the autapses can play an important role in the synchronous behaviour. To do so, we compute the order parameterto quantify synchronisation, the coefficient of variation in neural activity, firing rates and synaptic current inputs. In oursimulations, we observe that autapses can increase or decrease synchronous behaviour in neural networks with excitatorysynapses. However, when only inhibitory synapses are considered, synchronisation does not suffer significant alterations in thepresence of autapses. Interestingly, in networks with excitatory and inhibitory synapses, we show that excitatory autapses cangive rise to synchronous or desynchronous neural activity. Our results provide evidence how synchronous and desynchronousactivities can emerge in neural networks due to autapses and contribute to understanding further the relation between autapsesand neural synchronisation.The paper is organised as follows: In Sec. II, we introduce the neural network of AEIF neurons and the diagnostic tools thatwill be used, such as the order parameter for synchronisation, the coefficient of variation, the firing rates and synaptic currentinputs. In Sec. III, we present the results of our study concerning the effects of autapses in neural synchronisation, and in Sec.IV, we draw our conclusions. II. M ETHODS
A. The AEIF model with neural autapses and network configurations
The cortex comprises mainly excitatory pyramidal neurons and inhibitory interneurons [75]. Inhibitory neurons have arelatively higher firing rate than excitatory ones [76], [77], [78]. In the mammalian cortex, the firing pattern of excitatoryneurons corresponds to regular spiking [13], while inhibitory neurons exhibit fast spiking activities [79]. Furthermore, excitatoryneurons show adaptation properties in response to depolarising inputs and the inhibitory adaptation current is negligible ornonexistent [80], [81], [82], [83], [84], [85]. The fast spiking interneurons are the most common inhibitory neurons in thecortex [86].In the neural networks considered in this work, the dynamics of each neuron j , where j = , . . . , N , is given by the adaptiveexponential integrate-and-fire model. In this framework, N denotes the total number of neurons in the network. The AEIFmodel is able to reproduce different firing patterns, including regular and fast spiking [47]. The network dynamics is given bythe following set of coupled, nonlinear, ordinary differential equations C m dV j dt = − g L ( V j − E L ) + g L ∆ T exp (cid:18) V j − V T ∆ T (cid:19) − w j + I + I chem j ( t ) , τ w dw j dt = a j ( V j − E L ) − w j , (1) τ s dg j dt = − g j , where V j is the membrane potential, w j the adaptation current and g j the synaptic conductance of neuron j . k and j identifythe pre and postsynaptic neurons. When the membrane potential of neuron j is above the threshold V thres , i.e. when V j > V thres [87], the state variables are updated according to the rules V j → V r , w j → w j + b j , (2) g j → g j + g s , where g s assumes the value g aute for excitatory autapses, g e for synapses among excitatory neurons, g ei for synapses fromexcitatory to inhibitory neurons, g auti for inhibitory autapses, g i for synapses among inhibitory neurons and g ie for synapsesfrom inhibitory to excitatory neurons. We consider a neuron is excitatory (inhibitory) when it is connected to another neuronwith an excitatory (inhibitory) synapse. The initial conditions of V j are randomly distributed in the interval V j = [ − , − ] mV. The initial values of w j are randomly distributed in the interval w j = [ , ] pA for excitatory and w j = [ , ] pA forinhibitory neurons. We consider the initial value of g j equal to zero for all neurons. Table I summarises the description andvalues of the parameters used in the simulations. TABLE ID
ESCRIPTION AND VALUES OF THE PARAMETERS IN THE
AEIF
SYSTEM AND USED IN THE SIMULATIONS . V
ALUES FOR PARAMETERS FOREXCITATORY AND INHIBITORY CONNECTIONS ARE DENOTED BY • AND (cid:63) , RESPECTIVELY . Parameter Description Value N Number of AEIF neurons C m Membrane capacitance
200 pF g L Leak conductance
12 nS E L Leak reversal potential -70 mV I Constant input current
270 pA ∆ T Slope factor V T Potential threshold -50 mV τ w Adaptation time constant
300 ms τ s Synaptic time constant V r Reset potential -58 mV M exc jk Adjacency matrix elements M inh jk Adjacency matrix elements t ini Initial time in the analyses
10 s t fin Final time in the analyses
20 s a j Subthreshold adaptation [ . , . ] nS • (cid:63) b j Triggered adaptation
70 pA • (cid:63) V REV
Synaptic reversal potential V excREV = • V inhREV = −
80 mV (cid:63) g s Chemical conductances g e , g aute , g ei • g i , g auti , g ie (cid:63) g e Excitatory to excitatory [0,0.5] nS • g aute Excitatory autaptic [0,35] nS • g ei Excitatory to inhibitory [0,5] nS • g i Inhibitory to inhibitory [0,2] nS (cid:63) g auti Inhibitory autaptic [0,100] nS (cid:63) g ie Inhibitory to excitatory [0,3] nS (cid:63) d j Time delay d exc = . • d inh = . (cid:63) The synaptic current arriving at each neuron depends on specific parameters, including the connectivity encoded in theadjacency matrices M exc and M inh , i.e. in the excitatory and inhibitory connectivity matrices. In particular, the input current I chem j arriving at each neuron j , is calculated by I chem j ( t ) = I exc j ( t ) + I inh j ( t ) , (3)where I exc j ( t ) = I ee j ( t ) + I ei j ( t ) + I e , aut j ( t )= ( V excREV − V j ( t )) N ∑ k = M exc jk g k ( t − d exc ) (4)and I inh j ( t ) = I ii j ( t ) + I ie j ( t ) + I i , aut j ( t )= ( V inhREV − V j ( t )) N ∑ k = M inh jk g k ( t − d inh ) . (5)In this framework, the type of synapse (excitatory or inhibitory) depends on the synaptic reversal potential V REV . We consider V excREV = V inhREV = −
80 mV for inhibitory synapses. The time delay in the conductance of the pre-synapticneuron k ( g k ) assumes d exc = . d inh = . t = [ − d j , ] .The first N exc neurons are excitatory and the last N inh inhibitory. The connections that depart from excitatory and inhibitoryneurons are associated with the excitatory and inhibitory matrices, M exc and M inh , where each entry is denoted M exc jk and M inh jk ,respectively. These adjacency matrices are binary and have entries equal to 1 when there is a connection from neuron k toneuron j , or 0 otherwise, as shown in Fig. 1.We consider P exc =
80% excitatory and P inh =
20% inhibitory neural populations following [47], [88], where the numbersof excitatory and inhibitory neurons are given by N exc = P exc N and N inh = P inh N , respectively. The connectivity probabilitiesare set to p aute = p auti = .
25 for excitatory and inhibitory autapses, to p e = .
05 and p i = . neural population and to p ei = p ie = .
05 for connectivity among different neural populations [47]. The subscripts “e” and“i” stand for “excitatory” and “inhibitory”, respectively and the superscript “aut” stands for “autapses”. The terms p ei and p ie represent the probabilities of connections from excitatory to inhibitory and from inhibitory to excitatory neurons, respectively.The probabilities of excitatory and inhibitory autapses are defined by p aute = N aute N exc and p auti = N auti N inh , (6)where N aute and N auti are the number of autapses in the excitatory and inhibitory populations, respectively. For a network withonly excitatory (inhibitory) neurons, the number of excitatory (inhibitory) neurons is N exc = N ( N inh = N ). For connectionswithin the excitatory and inhibitory populations, the corresponding probabilities p e and p i are given by p e = N e N exc ( N exc − ) and p i = N i N inh ( N inh − ) , (7)where N e and N i are the number of synaptic connections in the excitatory and inhibitory populations, respectively. Forconnections among different populations, the corresponding probabilities are given by p ei = N ei N exc N inh and p ie = N ie N exc N inh , (8)where N ei and N ie are the number of synaptic connections from the excitatory to the inhibitory and from the inhibitory to theexcitatory populations, respectively. Therefore, when only one neural population is considered, p ei and p ie cannot be defined.The resulting 6 connectivity probabilities are represented in the connectivity matrix in Fig. 1, where k and j denote the pre-and post-synaptic neurons, respectively. Figure 1 shows the connections associated to probabilities: (a) in the same population( p e and p i ), (b) for autapses ( p aute and p auti ) and (c) among different populations ( p ei and p ie ). Fig. 1. (Colour online) Representation of the connections: (a) in the same population, (b) for autapses and (c) among different neural populations. Here, “pre”stands for “pre-synaptic” and “post” for “post-synaptic”. We note that we have used P exc =
80% excitatory (denoted red) and P inh =
20% inhibitory (denotedblue) neural populations which amounts to a total of N = Finally, we associate the conductances g e , g i , g aute , g auti , g ei and g ie to the corresponding connectivity probabilities discussedbefore. To solve the set of ordinary differential equations in system 1, we used the 4th order Runge-Kutta method with theintegration time-step equal to 10 − ms. B. Computation of neural synchronisation
Synchronous behaviour in neural networks can be quantified by means of the order parameter R [108] R ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N ∑ j = exp (cid:0) i ψ j ( t ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9)where R ( t ) is the amplitude of a centroid phase vector over time, i the imaginary unit, satisfying i = − |·| , the vector-normof the argument. The phase of each neuron j in time is obtained by means of ψ j ( t ) = π m + π t − t j , m t j , m + − t j , m , (10) where t j , m is the time of the m -th spike of neuron j , where t j , m < t < t j , m + [89]. We consider that spikes occur whenever V j > V thres [87]. R ( t ) takes values in [ , ] and, is equal to 0 for completely desynchronised neural activity and 1 for fullysynchronised neural behaviour. We compute the time-average order parameter R [90], given by R = t fin − t ini (cid:90) t fin t ini R ( t ) dt , (11)where ( t fin − t ini ) is the length of the time window [ t ini , t fin ] . Here, we have used t ini =
10 s and t fin =
20 s. Similarly, we calculatethe synchronisation of the non-autaptic neurons R non ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N non N non ∑ j = exp (cid:16) i ψ non j ( t ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (12)and autaptic neurons R aut ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N aut N aut ∑ j = exp (cid:16) i ψ aut j ( t ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13)where N non and N aut are the number of non-autaptic and autaptic neurons, respectively. In this context, ψ non j and ψ aut j are thephases of the non-autaptic and autaptic neuron j and both terms are computed using Eq. 10 for the times of spiking of thenon-autaptic and autaptic neurons, respectively. R non and R aut are then obtained according to Eq. 11. C. Mean coefficient of variation of interspike intervals
We calculate the interspike intervals of each neuron to obtain the mean coefficient of variation. In particular, the m -thinterspike interval of neuron j , ISI mj , is defined as the difference between two consecutive spikes,ISI mj = t j , m + − t j , m > , (14)where t j , m is the time of the m -th spike of neuron j . Using the mean value of ISI j over all m , ISI j and its standard deviation σ ISI j , we can compute the coefficient of variation (CV) of neuron j ,CV j = σ ISI j ISI j . (15)The average CV over all neurons in the network, CV, can then be computed byCV = N N ∑ j = CV j . (16)We use the value of CV to identify spikes whenever CV < . ≥ . D. Firing rates in neural populations
The mean firing-rate of all neurons in a network is computed by means of F = N ( t fin − t ini ) N ∑ j = (cid:32) (cid:90) t fin t ini δ ( t (cid:48) − t j ) dt (cid:48) (cid:33) , (17)where t j is the firing time of neuron j . In some occasions, we calculate the mean firing frequency of neurons with and withoutautapses, F aut = N autx ( t fin − t ini ) N autx ∑ j = (cid:32) (cid:90) t fin t ini δ ( t (cid:48) − t aut j ) dt (cid:48) (cid:33) (18)and F non = N nonx ( t fin − t ini ) N nonx ∑ j = (cid:32) (cid:90) t fin t ini δ ( t (cid:48) − t non j ) dt (cid:48) (cid:33) , (19)where N autx and N nonx are the number of neurons with and without autapses, and t aut j and t non j the firing times of neurons withand without autapses. The subscript “x” denotes the population of excitatory (“e”) or inhibitory (“i”) neurons. Similarly, we calculate the firing rate of excitatory and inhibitory neurons by means of F exc = N exc ( t fin − t ini ) N exc ∑ j = (cid:18) (cid:90) t fin t ini δ ( t (cid:48) − t exc j ) dt (cid:48) (cid:19) (20)and F inh = N inh ( t fin − t ini ) N inh ∑ j = (cid:18) (cid:90) t fin t ini δ ( t (cid:48) − t inh j ) dt (cid:48) (cid:19) , (21)where t exc j and t inh j are the firing times of the excitatory and inhibitory neurons, respectively. E. Synaptic current inputs
In our work, we calculate the mean instantaneous input I chem ( t ) and the time average of the synaptic input I s (pA) in thenetwork by I chem ( t ) = N N ∑ j = I chem j ( t ) (22)and I s = t fin − t ini (cid:90) t fin t ini I chem ( t ) dt , (23)respectively, where I chem j ( t ) is given by Eq. 3. In this respect, the values of I chem change over time due to excitatory andinhibitory inputs received by neuron j , where j = , . . . , N .III. R ESULTS AND D ISCUSSION
A. Network with excitatory neurons only
Networks with excitatory neurons were studied previously by Borges et al. [22] and Protachevicz et al. [26]. These studiesshowed that excitatory neurons can change firing patterns and improve neural synchronisation. Fardet et al. [92] and Yin etal. [93] reported that excitatory autapses with few milliseconds time delay can change neural activities from spikes to bursts.Wiles et. al [64] demonstrated that excitatory autaptic connections contribute more to bursting firing patterns than inhibitoryones.In Fig. 2, we consider a neural network with excitatory neurons only, where g e corresponds to the intensity of excitatorysynaptic conductance and g aute to the intensity of excitatory autaptic conductance. In our neural network, a neuron receives manyconnections from other neurons with small intensity of synaptic conductances. For the autaptic neurons, only one synapticcontact from a neuron to itself via a closed loop is considered. Due to this fact, to study the autaptic influence on the highsynchronous activities, we consider values of g aute greater than g e . Panel (a) shows a schematic representation of a neuralnetwork of excitatory neurons only with a single autapse represented by the closed loop with excitatory autaptic conductance g aute . Panels (b) to (d) give the mean order parameter in the parameter space g e × g aute . We see that excitatory autapses canincrease or reduce the synchronisation in a population of excitatory neurons when the intensity of the excitatory synapticconductance is small. In these panels, the circle ( g e = .
05 nS and g aute =
10 nS), triangle ( g e = .
05 nS and g aute =
31 nS),square ( g e = . g aute =
15 nS) and hexagon ( g e = . g aute =
22 nS) symbols indicate the values of the parametersshown in Fig. 3. We observe that desynchronous firing patterns as seen in Fig. 3(a) can become more synchronous, as it canbe seen in Fig. 3(b), due the increase of the excitatory autaptic conductance. On the other hand, the increase of the autapticconductance can decrease the level of synchronisation in the network, i.e. from high in Fig. 3(c) to low synchronous activitiesin Fig. 3(d). However, as shown in Fig. 2(d), the autaptic connections affect mainly the synchronisation of autaptic neurons.For a strong excitatory synaptic coupling ( g e ≥ . F ) and synaptic current ( I s ), respectively. Weverify that the excitatory autaptic neurons promote the increase of CV, F and I s in the network. In Fig. 2(e), we find thatboth synaptic and autaptic couplings can lead to burst activities, as reported by Borges et al. in [22] and Fardet et al. in [92].The burst and spike activities are characterised by CV ≥ . < . F non ) and with autapses ( F aut ), as well as of all neurons in the excitatory network ( F ). In Fig. 4(a), we Fig. 2. (Colour online) (a) Schematic representation of the neural network where g e is the intensity of excitatory synaptic conductance and g aute of theexcitatory autaptic conductance. Parameter space g e × g aute , where the colour bars correspond to (b) R , (c) R non , (d) R aut , (e) CV, (f) F and (g) I s . The rasterplots of the parameters indicated in (b), (c) and (d) (circle, square, triangle and hexagon) are shown in Fig. 3. The vertical and horizontal white, dash, lines inpanel (f) are used to vary g aute and g e in the computations in panels (a) and (b) in Fig. 4, respectively. The closed loop in panel (a) corresponds to an autapseof excitatory autaptic conductance g aute .Fig. 3. (Colour online) Raster plots for the neural network with excitatory neurons only. The values of the parameters g e and g aute are indicated in panels (b)to (d) in Fig. 2 by circle, triangle, square and hexagon symbols, respectively. The curly brackets in the upper right corner of the plots denote the autapticneurons considered. consider g e = . g aute , while in Fig. 4(b), we use g aute =
20 nS varying g e , as shown in Fig. 2(f) with white, dash,lines. We find that the autaptic connections increase the firing frequency of all neurons in the network and mainly those withautaptic connections. In our simulations, neurons with excitatory autapses exhibit the highest firing rate. Fig. 4. (Colour online) Plot of F non (black curve), F aut (red curve) and F (green curve) for (a) g e = . g aute and (b) g aute =
20 nS varying g e .Here, g aute and g e vary along the white, dash, lines in Fig. 2(f). B. Network with inhibitory neurons only
Synaptic inhibition regulates the level of neural activity and can prevent hyper excitability [94]. Studies have shown thatneural networks can exhibit synchronous activities due to inhibitory synapses [95], [96], [97], [98]. Here, we analyse theinfluence of inhibitory synapses and autapses by varying g i and g auti , as shown in Fig. 5(a). Panel (b) in the same figure showsthat inhibitory synapses and autapses do not give rise to the increase of neural synchronisation in the network. Actually, neuralsynchronisation due to inhibition is possible when it is considered together with other mechanisms related to neural interactions[99], e.g. with gap junctions associated to inhibitory synapses [68], [100], [101], [102], [103], [104], [105]. Fig. 5. (Colour online) (a) Schematic representation of an inhibitory neural population connected with inhibitory synapses and autapses. Parameter space g i × g auti , where the colour bars encode the values of (b) R , (c) CV, (d) I s , (e) F , (f) F non and (g) F aut . The vertical and horizontal black, dash, lines in panels(e) to (g) are used to vary the corresponding parameters in the computations in panels (a) and (b) in Fig. 6. The closed loop in panel (a) corresponds to anautapse of conductance intensity g auti . In our simulations, we do not observe that inhibitory interactions promote synchronisation in the network. Although thisis not surprising, it helps to identify the role of inhibitory autapses in neural synchronisation. Figure 5(c) shows that there isno change from spike to burst patterns, either. In Fig. 5(d), we verify that both inhibitory synapses and autapses increase theintensity of the mean negative synaptic current.In panel (e) in Fig. 5, we see that inhibitory synapses contribute to the decrease of F , while panels (f) and (g) in show themean firing rate for non-autaptic neurons, i.e. neurons without autapses ( F non ) and for autaptic neurons ( F aut ), respectively.The autapses reduce the firing-rate of the autaptic neurons, what can lead to an increase of the firing rate of the non-autapticneurons. This can be better observed in Fig. 6(a), which shows the values of F non , F aut and F as a function of g auti for g i = . Fig. 6. (Colour online) Plot of F non (black line), F aut (red line) and F (green line) for (a) g i = . g auti and (b) g auti =
10 nS varying g i , indicatedin Fig. 5 by the black, dash, lines. nS. Figure 6(b) shows the mean firing rates as a function of g i for g auti =
10 nS. The neurons with inhibitory autapses havelower firing rates.
C. Network with a mix of excitatory and inhibitory neurons
Desynchronous neural activities in balanced excitatory/inhibitory regimes have been reported in [85], [25]. Based on theseresults, here we study different combinations of g e , g i , g aute and g auti values in the parameter space g ei × g ie (see Fig. 7). Theexistence of synchronous and desynchronous activities depend on the values of these parameters which are related to theconductances. We focus on a set of parameters for which synchronous activities appear. Firstly, we consider g e = . g i = Fig. 7. (Colour online) (a) Schematic representation of a neural network with a mix of excitatory and inhibitory neurons without autapses. Parameter spaces g ei × g ie for g e = . g i = R , (c) F , (d) F exc and (e) F inh . The circle, square and triangle symbols in (b)represent the values of the parameters considered in the computation of the raster plots shown in the right side. The blue and red points in the raster plotsindicate the firing of the inhibitory and excitatory neurons over time, respectively. Figure 7(a) shows a schematic representation of excitatory (red circles) and inhibitory (blue circles) neurons, where g ei ( g ie ) correspond to the conductance from excitatory to inhibitory (from inhibitory to excitatory) neurons in the absence ofautapses. Panel (b) presents the mean order parameter ( R ) and the circle, square and triangle symbols indicate the values ofthe parameters considered in the computation of the raster plots shown in the right hand-side. The values of the conductancesused to compute the raster plots are given by g ei = . g ie = . g ei = . g ie = . g ei = . g ie = . be suppressed by means of inhibitory to excitatory or excitatory to inhibitory connection heterogeneity. Here, we observe thata minimal interaction between the excitatory and inhibitory neurons is required to suppress high synchronous patterns. In Fig.7(c), we verify that F decreases when g ie increases. Panels (d) and (e) show that F exc and F ini can decrease when g ie increases.In addition, F exc decreases and F ini increases when g ei increases. When the neural populations are uncoupled ( g ei = g ie = g ei × g ie for g aute =
30 nS, where the colour bar correspondsto R . The white solid line in the parameter space indicates the transition from desynchronous to synchronous behaviour in thenetwork without excitatory autaptic conductance ( g aute = g ei and g ie . Fig. 8. (Colour online) (a) Schematic representation of a neural network with a mix of excitatory and inhibitory neurons with excitatory autapses. Parameterspaces g ei × g ie for g e = . g i = g aute =
30 nS, where the colour bars correspond to (b) R , (c) F , (d) F exc and (e) F inh . The circle, square andtriangle symbols in (b) represent the values of the parameters considered in the computation of the raster plots shown in the right side. The blue and redpoints in the raster plots indicate the firing of the inhibitory and excitatory neurons over time, respectively. The curly brackets in the upper left corner of theplots denote the autaptic neurons considered. IV. C
ONCLUSIONS
In this paper, we investigated the influence of autapses on neural synchronisation in networks of coupled adaptive exponentialintegrate-and-fire neurons. Depending on the parameters of the system, the AEIF model exhibits spike or burst activity. In oursimulations, we considered neurons randomly connected with chemical synapses in the absence or presence of autapses.We verified that the type of synaptic connectivity plays a different role in the dynamics in the neural network, especiallywith regard to synchronisation. It has been reported that excitatory synapses promote synchronisation and firing patterntransitions. In our simulations, we found that excitatory autapses can generate firing pattern transitions for low excitatorysynaptic conductances. The excitatory autaptic connections can promote desynchronisation of all neurons or only of the autapticones in a network with neurons initially synchronised. The excitatory autapses can also increase the firing rate of all neurons.In a network with only inhibitory synapses, we did not observe inhibitory synapses and autapses promoting synchronisation.We saw a reduction and increase of the firing rate of the autaptic and non-autaptic neurons, respectively, due to inhibitoryautapses.Finally, in a network with a mix of excitatory and inhibitory neurons, we saw that the interactions among the populationsare essential to avoid high synchronous behaviour. The excitatory to inhibitory synaptic connectivities promote the increase(decrease) of the firing rate of the inhibitory (excitatory) populations. On the other hand, the inhibitory to excitatory synapticconnectivities give rise to the decrease of the firing rate of both populations. We observed that the excitatory autapses canreduce the synchronous activities, as well as induce neural synchronisation. For small conductances, excitatory autapses cannot change synchronisation significantly. Consequently, our results provide evidence on the synchronous and desynchronousactivities that emerge in random neural networks with chemical, inhibitory and excitatory, synapses where some neurons areequipped with autapses.In a more general context, the role of network structure upon synchronicity in networks with delayed coupling and delayedfeedback was studied, and very general classifications of the network topology for large delay were given by Flunkert et al. [106], [107], e.g., it was shown that adding time-delayed feedback loops to a unidirectionally coupled ring enables stabilisationof the chaotic synchronisation, since it changes the network class. We believe that the absence or presence of autapses hassimilar effects upon synchronisation. In future works, we plan to compute the master stability function of networks withautapses to compare with the stability of synchronisation in delay-coupled networks.C ONFLICT OF I NTEREST S TATEMENT
The authors declare that there is no conflict of interest.A
UTHOR C ONTRIBUTIONS
All authors discussed the results and contributed to the final version of the manuscript.F
UNDING
This study was possible by partial financial support provided by the following Brazilian government agencies: Fundac¸ ˜aoArauc´aria, National Council for Scientific and Technological Development (CNPq), Coordenac¸ ˜ao de Aperfeic¸oamento dePessoal de N´ıvel Superior-Brasil (CAPES) and S˜ao Paulo Research Foundation (FAPESP) (2020/04624-2). We also wish tothank Newton Fund, IRTG 1740/TRP 2015/50122-0 funded by DFG/FAPESP and the RF Government Grant 075-15-2019-1885.Support from Russian Ministry of Science and Education “Digital biodesign and personalised healthcare”.R
EFERENCES[1] Tatii, R., Haley, M. S., Swanson, O., Tselha, T., Maffei, A. (2018). Neurophysiology and regulation of the balance between excitation and inhibitionin neocortical circuits.
Biological Psychiatry
15, 821-831.[2] Zhou, S., Yu, Y. (2018). Synaptic E-I balance underlies efficient neural coding.
Frontiers in Neuroscience
12, 46.[3] Drongelen, W., Lee, H. C., Hereld, M., Chen, Z., Elsen, F. P., Stevens, R. L. (2005). Emergent epileptiform activity in neural networks with weakexcitatory synapses.
IEEE Transactions on neural systems and rehabilitation engineering
13, 236-242.[4] Avoli, M., Curtis, M., Gnatkovsky, V., Gotman, J., K¨ohling, R. L´evesque, M., Manseau, F., Shiri, Z., Williams, S. (2016). Specific imbalance ofexcitatory/inhibitory signaling establishes seizure onset pattern in temporal lobe epilepsy.
Journal of Neurophysiology
Frontiers inComputational Neuroscience
12, 105.[6] Sporns, O. (2013). Structure and function of complex brain networks.
Dialogues in clinical neuroscience
Frontiers in Neural Circuits , 8, 112.[8] Sporns, O. Connectome networks: from cells to systems. (2016). In: Kennedy H, Van Essen DC, Christen Y, editors. Micro-, meso- and macro-connectomics of the brain.
Springer
Trends in CognitiveSciences
24, 302-315.[10] Sporns, O. (2012). Discovering the human connectome.
MIT press , 63-84.[11] Feng, Z., Zheng, M., Chen, X., Zhang, M. (2018). neural synapses: microscale signal processing machineries formed by phase separation?
Biochemistry
57, 2530-2539.[12] Bittner, S. R., Williamson, R. C., Snyder, A. C., Litwin-Kumar, A., Doiron, B., Chase, S. M., Smith, M. A., Yu, B. M. (2017). Population activitystructure of excitatory and inhibitory neurons.
PLOS ONE
The Journal of Neuroscience
Journal of Neurophysiology
Neural Computation
15 (3), 509-38.[16] Han, F., Gu, X., Wang, Z., Fan, H., Cao, J., Lu, Q. (2018). Global firing rate contrast enhancement in E/I neural networks by recurrent synchronizedinhibition.
Chaos
28, 106324.[17] Hayakawa, T., Fukai, T. (2020). Spontaneous and stimulus-induced coherent states of critically balanced neural networks.
Physical Review Research
Proceedingsof the National Academy of Sciences , 115, 3464-3469.[19] Mahmud, M., Vassanelli, S. (2016). Differential modulation of excitatory and inhibitory neurons during periodic stimulation.
Frontiers in Neuroscience
10, 62.[20] Deco, G., Ponce-Alvarez, A., Hagmann, P., Romani, G. L., Mantini, D., Corbetta, M. (2014). How local excitatory-inhibition ratio impact the wholebrain dynamics.
The Journal of Neuroscience
34, 7886-7998.[21] Mejias, J. F., Murray, J. D., Kennedy, H., Wang, X.-J. (2016). Feedforward and feedback frequency-dependent interactions in a large-scale laminarnetwork of the primate cortex.
Science Advances
2, e1601335.[22] Borges, F. S., Protachevicz, P. R., Lameu, E. L., Bonetti, R. C., Iarosz, K. C., Caldas, I. L., Baptista, M. S., Batista., A. M. (2017). Synchronised firingpatterns in a random network of adaptive exponential integrate-and-fire neuron model.
Neural Networks
90, 1-7.[23] Breakspear, M., Terry, J. R., Friston, K. J. (2003). Modulation of excitatory synaptic coupling facilitates synchronization and complex dynamics in anonlinear model of neural dynamics.
Neurocomputing
Frontiers in Computational Neuroscience
10, 109.[25] Ostojic, S. (2014). Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons.
Nature Neuroscience
17, 594-600.[26] Protachevicz, P. R., Borges, F. S., Lameu, E. L., Ji, P., Iarosz, K. C., Kihara, A. H., Caldas, I. L, Szezech Jr., J. D., Baptista, M. S., Macau, E. E. N.,Antonopoulos, C. G., Batista, A. M., Kurths, J. (2019). Bistable firing pattern in a neural network model.
Frontiers in Computational Neuroscience
Plos One [28] Protachevicz, P. R., Borges, F. S., Iarosz, K. C., Baptista, M. S., Lameu, E. L., Hansen, M., Caldas, I. L., Szezech Jr., J. D., Batista, A. M., Kurths, J.(2020). Influence of delayed conductance on neural synchronisation. Frontiers in Physiology
11, 1053.[29] Ge, P., Cao, H. (2019). Synchronization of Rulkov neuron networks coupled by excitatory and inhibitory chemical synapses.
Chaos
29, 023129.[30] Bondarenko, V. E., Chay, T. R. (1998). Desynchronization and synchronization processes in a randomly coupled ensemble of neurons.
Physical ReviewE
58, 8036-8040.[31] Gray, R. T., Robinson, P. A. (2008). Stability and synchronization of random brain networks with a distribution of connection strengths.
Neurocomputing
71, 1373-1387.[32] Antonopoulos, C. G., Srivastava, S., Pinto, S. E. D. S., Baptista, M. S. (2015). Do brain networks evolve by maximizing their information flow capacity?
PLOS Computational Biology
European Physical Journal: SpecialTopics
Scientific Reports
6, 19845.[35] Kim, S.-Y., Lim, W. (2013). Sparsely-synchronized brain rhythm in a small-world neural network.
Journal of the Korean Physical Society
63, 104-113.[36] Li, C., Zheng, Q. (2010). Synchronization of the small-world neural network with unreliable synapses.
Physical Biology
7, 036010.[37] Qu, J., Wang, R., Yan, C., Du, Y. (2014). Oscillations and synchrony in a cortical neural network.
Cognitive Neurodynamics
8, 157-166.[38] Santos, M. S., Protachevicz, P. R., Iarosz, K. C., Caldas, I. L, Viana, R. L., Borges, F. S, Ren, H.-P., Szezech Jr.,J. D., Batista, A. M., Grebogi, C.(2019). Spike-burst chimera states in an adaptive exponential integrate-and-fire neural network.
Chaos
29, 043106.[39] Wang, Q. Y., Lu, Q. S., Chen, G. R. (2007). Ordered bursting synchronized and complex wave propagation in a ring neural network.
Physica A
Chaos
27, 047402.[41] Wang, Q., Chen, G., Perc, M. (2011). Synchronous bursts on scale-free neural networks with attractive and repulsive coupling.
PLOS ONE
6, e15851.[42] Bekkers, J. M. (1998). Neurophysiology: Are autapses prodigal synapses?
Current Biology
8, R52-R55.[43] Pouzat, C., Marty, A. (1998). Autaptic inhibitory currents recorder from interneurones in rat cerebellar slices.
Journal of Physiology
Chinese Physical B
24, 128709.[45] Bekkers, J. M. (2009). Synaptic Transmission: Excitatory autapses find a function?
Current Biology
19, R296.[46] van der Loos, H. Glaser, E. M. (1972). Autapses in neocortex cerebri: synaptic between a pyramidal cell’s axon and its own dendrites.
Brain Research
48, 355-360.[47] di Volo, M., Romagnoni, A., Capone, C. Destexhe, A. (2019). Biologically realistic mean-field models of conductance-based networks of spikingneurons with adaptation.
Neural Computation
Science China Technological Sciences
59, 364-370.[49] Bacci, A., Huguenard, J. R., Prince, D.A. (2003). Functional autaptic neurotransmission in fast-spiking interneurons: A novel form of feedback inhibitionin the neocortex. Journal of Neuroscience 23, 859-866.[50] Tam´as, G., Buhl, E. H., Somogyi, P. (1997). Massive autaptic self-innervation of GABAergic neurons in cat visual cortex.
The Journal of Neuroscience
17, 6352-6364.[51] Saada-Madar, R., Miller, N., Susswein, A. J. (2012). Autaptic Muscarinic self-excitation and nitrergic self-inhibition in neurons initiating Aplysiafeeding are revealed when the neurons are cultured in isolation.
Journal of Molecular Histology
43, 431-436.[52] Suga, K. (2014). Isoproterenol facilitates GABAergic autapses in fast-spiking cells of rat insular cortex.
Journal of Oral Science
56, 41-47.[53] Szegedi, V., Paizs, M., Baka, J.Barz´o, P., Moln´ar, G., Tamas, G., Lamsa, K. (2020). Robust perisomatic GABAergic self-innervation inhibits basketcells in the human and mouse supragranullar neocortex. eLife
9, e51691.[54] Wyart, C., Cocco, S., Bourdieu, L., L´eger, J.-F., Herr, C., Chatenay, D. (2005). Dynamics of excitatory synaptic components in sustained firing at lowrates.
The Journal of Neuroscience
93, 3370-3380.[55] Valente, P., Orlando, M., Raimondi, A., Benfenati, F., Baldelli, P. (2016). Fine tuning of synaptic plasticity and filtering by GABA released fromhippocampal autaptic granule cells.
Cerebral Cortex
26, 1149-1167.[56] Wang, C., Guo, S., Xu, Y., Ma, J., Tang, J., Alzahrani, F., Hobiny, A. (2017). Formation of autapse connected to neuron and it biological function.
Research Article
Nonlinear Dynamics
Chinese Physics B
24, 128709.[59] Herrmann, C. S., Klaus, A. (2004). Autapse turns neuron into oscillator.
International Journal of Bifurcation and Chaos
14, 623-633.[60] Jia, B. (2018). Negative feedback mediated by fast inhibitory autapse enhances neural oscillations near a hopf bifurcation point.
International Journalof Bifurcation and Chaos
28, 1850030.[61] Kim, Y. (2019). Autaptic effects on synchronization and phase response curves of neurons with a chemical synapse.
The Korean Physical Society
ScienceChina Technological Sciences
57, 936-946.[63] Zhao, Z., Gu, H. (2017). Transitions between classes of neural excitability and bifurcations induced by autapse.
Scientific Reports
7, 6760.[64] Wiles, L., Gu. S., Pasqueletti, F., Parvesse, B., Gabrieli, D. Basset, D. S., Mean, D. F. (2017). Autaptic connections shitf network excitability andbursting.
Scientific Reports
7, 44006.[65] Ke,W., He, Q., Shu, Y. (2019). Functional self-excitatory autapses (auto-synapses) on neocortical pyramidal cells.
Neuroscience bulletin
35, 1106-1109.[66] Bekkers, J. M. (2003). Synaptic Transmission: Functional autapses in the cortex.
Current Biology
13, R443-435.[67] Qin, H., Wu, Y., Wang, C., Ma, J. (2014). Emitting waves from defects in network with autapses.
Communications in Nonlinear Science and NumericalSimulation
23, 164-174.[68] Guo, D., Wang, Q., Perc, M. (2012). Complex synchronous behavior in interneural networks with delayed inhibitory and fast electrical synapses.
Physical Review E
85, 061905.[69] Guo, D., Wu, S., Chen, M., Perc, M., Zhang, Y., Ma, J., Cui, Y., Xu, P., Xia, Y., Yao, D. (2016). Regulation of irregular neural firing by autaptictransmission.
Scientific Reports
6, 26096.[70] Wang, H., Wang, L., Chen, Y., Chen, Y. (2014). Effect of autaptic activity on the response of a Hodgkin-Huxley neuron.
Chaos
24, 033122.[71] Deleuze, C., Bhumbra, G. S., Pazienti, A., Lourenco, J., Mailhes, C., Aguirre, A., Beato, M., Bacci , A. (2019). Strong preference for autapticself-connectivity of neocortical PV interneurons facilitates their tuning to γ -oscillations. PLOS Biology
17, e3000419.[72] Seung, H. S., Lee, D. D., Reis, B. Y., Tank, D. W. (2000). The autapse: A simple illustration of short-term analog memory storage by tuned synapticfeedback.
Journal of Computational Neuroscience
9, 171-185.[73] Brette, R., Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neural activity.
Journal of Neurophysiology
94, 3637-3642. [74] Ladenbauer, J., Lehnert, J., Rankoohi, H., Dahms, T., Sch¨oll, E., Obermayer, K. (2013). Adaptation controls synchrony and cluster states of coupledthreshold-model neurons. Physical Review E
88, 042713.[75] Atencio, C. A., Schreiner, C. E. (2008). Spectrotemporal processing differences between auditory cortical fast-spiking and regular-spiking neurons.
TheJournal of Neuroscience γ -aminobutyrate-containing neurons andpyramidal neurons in prefrontal cortex. PNAS
1, 32-36.[78] Baeg, E. H., Kim, Y. B., Jang, J., Kim, H. T., Mook-Jung, I., Jung, M. W. (2001). Fast spiking and regular spiking neural correlates of fear conditioningin the medial prefrontal cortex of the rat.
Cerebral Cortex
11, 441-451.[79] Wang, B., Ke, W., Guang, J., Chen G., Yin, L., Deng, S., He, Q., Liu, Y., He, T., Zheng, R., Jiang, Y., Zhang, X., Li, T., Luan, G., Lu, H. D.,Zhang, M., Zhang, X. Shu, Y. (2016). Firing frequency maxima of fast-spiking neurons in human, monkey and mouse neocortex.
Frontiers in CellularNeuroscience
Journal of Neurophysiology
VisualNeuroscience
15, 979-993.[82] Hensch, T. K., Fagiolini, M. (2004). Excitatory-inhibitory balance, synapses, circuits, systems.
Springer Science+Bussiness Media , New York, 155-172.[83] Destexhe, A. (2009). Self-sustained asynchronous irregular states and up-down states in thalamic, cortical and thalamocortical networks of nonlinearintegrate-and-fire neurons.
Journal of Computational Neuroscience
Proceeingsof 4th International Coneference on NeuroRehabilitation
21, 58-63.[85] Borges, F. S., Protachevicz, P. R., Pena, R. F. O., Lameu, E. L., Higa, G. S. V., Kihara, A. H., Matias, F. S., Antonopoulos, C. G., de Pasquale, R.,Roque, A. C., Iarosz, K. C., Ji, P., Batista, A. M. (2020). Self-sustained activity of low firing rate in balanced networks.
Physica A
Proceedingsof the National Academy of Sciences
Biological Cybernetics
99, 335-347.[88] Noback, C. R., Strominger, N. L., Demarest, R. J., Ruggiero, D. A. (2005).
The Human Nervous System: Structure and Function (Sixth ed.). Totowa,NJ: Humana Press.[89] Rosenblum, M. G., Pikowsky, A. S., Kurths, J. (1997). From phase to lag synchronization in coupled chaotic oscillators.
Physical Review Letters
Physica A
PhysiologicalMeasurement
Frontiers in Neuroscience
2, 1-14.[93] Yin, L., Zheng, R., Ke, W., He, Q., Zhang, Y., Li, J., Wang, B. Long, Y.-S., Rasch, M. J., Li, T., Luan, G., Shu, Y. (2018). Autapses enhance burstingand coincidence detection in neocortical pyramidal cells.
Nature Communications
9, 4890.[94] Fr¨ohlich, F. (2016). Microcircuits of the Neocortex. Network Neuroscience.
Academic Press
1, 85-95.[95] Vreeswijk, C. V., Abbot, L. F., Ermentrout, G. B. (1994). When inhibition not excitation synchronizes neural firing.
Journal of ComputationalNeuroscience
1, 313-321.[96] Elson, R. C., Selverston, A. I., Abarbanel, H. D., Rabinovich, M. I. (2002). Inhibitory synchronization of bursting in biological neurons: dependenceon synaptic time constant.
Journal of Neurophysiology
Europhysics Letters
ScientificReports
8, 11431.[99] Bartos, M., Imre. V., Michael, F., Axel, M., Hannah, M., Geiger, J. R. P., Peter, J. (2002). Fast synaptic inhibition promotes synchronized gammaoscillations in hippocampal interneuron networks.
Proceedings of the National Academy of Sciences
The Journal ofNeurophysiology , 85(4), 1543-1551.[101] Beierlein, M., Gibson, J. R., Connors, B. W. (2000). A network of electrically coupled interneurons drives synchronized inhibition in neocortex.
NatureNeuroscience
3, 904-910.[102] Kopell, N., Ermentrout, B. (2004) Chemical and electrical synapses perform complementary roles in the synchronization of interneural networks.
Proceedings of the National Academy of Sciences
Nature Reviews
Frontiers in ComputationalNeuroscience
Phylosophical Transaction A
Physical Review Letters