arXiv:1906.03066v1 [physics.bio-ph] 7 Jun 2019 hte hypeetaykn fpatct 1] Even [12]. plasticity of known kind not any is present it and rarely they cul- diagrams are whether hippocampal circuit connections cortical and autaptic in However, [4] included neocortex 11]. axo- [10, the 9], axo-axonic, in tures [8, found be been have can cerebellum and autapses dendo-dendritic a or Inhibitory to dendritic architec- cultures cortical [6]. experimental the ture in in struc- present relevant appears self-innervation not massive only from that connections changed tures these have the of neurobiologists from status then, neurons the Since pyramidal in [7]. ago neocortex years five forty than of net- rhythm the of firing [3–6]. properties work the synchronization the both and influence neuron the domain, to somato-dendritic shown own its been to have from neuron a synapses of be autaptic axon could of the which presence autapses), the flow particular, (or In connections information [2]. the network modulate the in neuron net- struc- ex- each intrinsic effective the the of that and the citability motifs means neuronal of This of connectivity blocks brain. tural the building in as dynamics work viewed oscillations be coherent can producing neu- microcircuits the level, At ronal [1]. patterns present connectivity can functional connectivity different anatomical same the that quires ∗ fernanda@fis.ufal.br h rtosraino uasswsrpre more reported was autapses of observation first The re- networks neuronal among communication Flexible niioyatpemdae niiae synchronization anticipated mediates autapse Inhibitory 3 ASnmes 71.n 71.l 87.19.lm 87.19.ll, 87.18.Sn, numbers: PACS underly mechanism the transition. t be DS-AS of could the S dynamics which the internal uncoupled, than are faster faster neurons a is promotes model Receiver autapse when inhi the inhibitory the which robust of in be values regime to large occu phase-drift shown extremely typically For is AS range. phenomenon physiological to The pla (DS) biologically increased. this synchronization is In delayed dela AS. negative exhibit usual the to the as order acts in and systems GA neuron ical The Receiver AS. the present auta may of inhibitory architecture, dynamics an cortical of the presence in the present pro in was model inhibition two-neuron delayed a o the that to where shown synapses was chemical AS dard Recently, self-feedback. negative delayed grto a xii niiae ycrnzto A)if (AS) synchronization anticipated exhibit can figuration ope ytm ru,Fcla eIgne´ayCeca A Ciencias Ingenier´ıa y de Facultad Group, Systems Complex w dnia uooosdnmclssesunidirectiona systems dynamical autonomous identical Two 2 .INTRODUCTION I. 1 eatmnod nomaiae au,Hsia tlaod Italiano Hospital Salud, Inform´atica en de Departamento nttt eFıia nvriaeFdrld lga,Mac Alagoas, de Federal F´ısica, Universidade de Instituto aclA Pinto, A. Marcel vnd Monseor Avenida 19AB iddAtooad unsArs Argentina. Aires, Buenos Aut´onoma de Ciudad C1199ABB, 1 laodlPril 245 a ods atao Chile. Santiago, Condes, Las 12.455, Portillo del Alvaro ´ sad .Rosso, A. Osvaldo ecie ytefloigsto equations: system of shown a set of been following solution the counter-intuitive has by but AS antici- described stable a present be [13]. to to (AS) order synchronization in pated unidirectionally systems by dynamical required coupled self-feedback delayed ative as act could they that [10]. decades lacking, function for still self-inhibitory speculated is autapses been the has of it role functional the though f yaia system, dynamical n seto hsslto sta natime a in that theoreti- strik- is of The solution this variety studies. of a [20–22] aspect ing in experimental and verified [13–19] been cal has and nization R t htA srbs gis os 2]adspike-timing and show [26] to extended noise been against has robust result interneu- is later an AS The by in that in mediated as synapses [25]. was well chemical ron as self-feedback by [24], the coupled which motif self-feedback the three-neuron as a and Rulkov delay term two synaptic between memory of verified a presence been the also in modeled has neurons be map-based AS could 1. they that, Eq. stimulus, way by external a an FitzHugh- such two from in with apart coupled [23] diffusively al. neurons et Ciszak Nagumo by done was model ae term layed R ,2 3 2, 1, ( + ( S eew rps htatpe ol c steneg- the as act could autapses that propose we Here h rtvrfiaino niiaini neuronal a in anticipation of verification first The t rdcstesaeo h Sender the of state the predicts savco ucinta ecie h autonomous the describes that function vector a is ) = ) t d . ioyatpetesse nege oa to undergoes system the autapse bitory n ennaS Matias S. Fernanda and sbeseai,asot rniinfrom transition smooth a scenario, usible h eevrnuo R lorcie a receives also (R) neuron Receiver the efe-unn eevrwe h two the when Receiver free-running he cri he-ernmtfwt stan- with motif three-neuron a in ccur S ie ya nenuo.Hr eshow we Here interneuron. an by vided ne.Frhroe eso htthe that show we Furthermore, ender. s,wihi asv self-innervation massive a is which pse, Ari uas euae h internal the regulates autapse BAergic ( e effebc eurdb dynam- by required self-feedback yed R swe h niioyconductance inhibitory the when rs l ope nasne-eevrcon- sender-receiver a in coupled lly t S ˙ n niiae ycrnzto and synchronization anticipated ing ˙ R lcds nvria elsAndes, los de Universidad plicadas, + io lga 77-7 Brazil 57072-970 ei´o, Alagoas unsAires Buenos e ( = = t t − d aaeesaevre ihna within varied are parameters ,caatrzsteatcptdsynchro- anticipated the characterizes ), f f t ( ( d R S stesl-edak[3.Tesolution The [13]. self-feedback the is ) K ewe ope neurons coupled between ( ( t t )) )+ )) steculn arxadtede- the and matrix coupling the is , K & [ S CONICET, 1, ( t ) ∗ − R S ( t naftr time future a in − t t d h Receiver the )] . (1) 2
spiking activity of the Sender (S) neuron is described by the equations bellow:
2 v˙S = 0.04vS +5vS + 140 − uS + I (2) u˙S = a(bvS − uS). (3)
FIG. 1: (Color online) Neuronal motif: Sender (S) and The Receiver (R) neuron has similar equations but with Receiver (R) neurons unidirectionally connected by an exci- two extra terms which accounts for the excitatory (E) tatory chemical synapse. The Receiver neuron presents an inhibitory chemical autapse which is a dynamical version of and self-inhibitory (I) synaptic currents: the negative delayed self-feedback in Eq. 1 . 2 v˙R = 0.04vR +5vR + 140 − uR + I + + gErE(EE − vS )+ gI rI (EI − vR) (4) dependent plasticity [27]. AS has also been verified in u˙R = a(bvR − uR). (5) cortical-like populations in the presence of excitatory and inhibitory neurons [28], which could explain a posi- For both neurons the reset condition is given by: if vi ≥ tive and unidirectional Granger causality and a negative 30 mV, then vi ←− c and ui ←− ui +d. The variable vi is phase difference between cortical areas of a non-human the equivalent of the membrane potential of the neuron primate [28–31]. and ui represents a membrane recovery variable. The Furthermore, it has been shown that the internal dy- subscript i indexes each neuron i = R,S. We employ namics of the Receiver neuron may control the rela- the same set of parameters for both neurons: a = 0.02, tive phase between the Sender and the Receiver neurons b = 0.2, c = −65 and d = 8. I is the external constant if they are synchronized [19]. In fact, anticipation in current which determines the neuronal firing rate. spike synchronization has been shown between two uni- The R neuron is subject to one excitatory synapses directionally coupled nonidentical chaotic neurons when from the S neuron mediated by AMPA receptors with the mean frequency of the free post-synaptic neuron is synaptic conductance gE. R is also subject to the in- greater than the pre-synaptic one [32]. AS has also been hibitory autapse mediated by GABAA receptors with verified between two Hodgkin-Huxley neurons coupled by autaptic conductance gI . The AMPA and GABAA re- a nonlinear excitatory synapse in which the depolariza- versal potentials are respectively EE = 0 mV and EI = tion levels (but not the synaptic conductance) determine −80 mV. The fraction of bound (i.e. open) synaptic re- the phase differences [33]. More recently, AS has also ceptors rj is modeled by a first-order kinetic dynamics: been verified in population neural models representing a microcircuit of sthe ongbird brain involved in song pro- r˙j = αj [T ](1 − rj ) − βj rj , (6) duction in which the Receiver dynamics is faster than the Senders [34]. where αj and βj are rate constants. The index j repre- Here we show that a two-neuron motif with an in- sents the excitatory synapse and the inhibitory autapse hibitory autapse in the post-synaptic neuron can present j = E, I.[T ] is the neurotransmitter concentration in an anticipatory regime. The autapse can act as a dy- the synaptic cleft. In its simplest model it is an instan- namical self-feedback and it can regulate the internal taneous function of the pre-synaptic potential vpre [36]: dynamics of the Receiver neuron. We employ numeri- cal simulations to show that GABAergic autapses can Tmax [T ](vpre)= − − . (7) be a biologically plausible mechanism for AS in neuronal 1+ e[ (vpre Vp)/Kp] motifs. For very small values of the inhibitory conduc- −1 tance the neurons synchronize with the usual pre-post In our model Tmax = 1 mM , Kp = 5 mV, Vp = 2 mV. order which is called the delayed synchronization regime Unless otherwise stated, the rate constants are αE = −1 −1 −1 −1 −1 (DS). As we increase the inhibition the system presents 1.1 mM ms , βE =0.19 ms , αI =5.0 mM ms , −1 a conductance-induced DS-AS transition. In Sec. II we and βI =0.30 ms similarly to the ones in Refs. [25, 36]. describe our simple motif as well as the neuronal and However, these values depend on a number of different synaptic models. In Sec. III, we report our results, show- factors and can vary significantly [37, 38]. ing that AS can be mediated by an inhibitory autapse in physiological regions of parameter space. Concluding remarks and briefly discussion of the significance of our III. RESULTS findings for neuroscience are presented in Sec. IV. Initially, we describe our results for the scenario where both neurons are subjected to a constant current I > II. MODEL 5 pA. Unless otherwise stated we keep the excitatory con- ductance fixed as gE =0.3 nS which is larger enough to Our motif is composed of two Izhikevich neuron mod- promote a phase-locking regime between the two neurons els [35] and chemical synapses as shown in Fig.1. The when there is no autapse. For different sets of inhibitory 3
(a) (see middle panels in Fig. 2a and b, for gI = 1.0 nS). If
30 τi does not converge to a fixed value, the system is in a 0 DS S phase-drift (PD) regime [39] (see bottom panels Fig. 2a -30 R and b, for gI =2.0 nS).
V (mV) -60
30 0 AS 0.0 -30
V (mV) -60 -0.1 30 0 PD -30 -0.2 / T V (mV) -60 I = 5 pA τ 394.08 394.12 394.16 394.20 -0.3 I = 6 pA t (s) I = 7 pA (b) I = 8 pA -0.4 I = 9 pA I =10 pA -0.5 0 1 2 3 gI (nS)
FIG. 3: (Color online) Transition from delayed to anticipated synchronization regime. Spiking time dif- ference normalized by the period of the phase-locking τ/T as a function of inhibitory conductance gI for different values of external current I. Positive values of τ characterizes DS, whereas τ < 0 represents the AS regime. The stars mark the end of the phase-locking regime and the beginning of the phase-drift (PD). In PD the value of τi in each cycle does not converge to a fixed value τ as shown in Fig. 2(b). FIG. 2: (Color online) Characterizing the phase- locking and phase-drift regimes. (a) Time series of the Sender and the Receiver neuron after a transient time for different values of the autaptic inhibition gI and fixed external current I = 10 pA. (b) The spike-timing difference τi between A. The conductance induced transition from DS to S and R as a function of the i-th cycle. A pre-post firing order AS characterizes the usual delayed (DS) synchronization regime (upper panels, gI = 0.15 nS). The anticipated synchronization The spike-timing difference τ is a continuous and regime (AS) occurs when the Receiver neuron fires before the smooth function of gI . The transition from delayed to Sender (middle panels, gI = 1.0 nS). When neurons fire with anticipated regime through a zero-lag regime may be in- different frequencies, the spike timing difference changes ev- ery cycle and the neurons are in a phase-drift (PD) regime duced by the inhibitory autaptic conductance as shown (bottom panels, gI = 2.0 nS). in Fig. 3. To compare the transition for different values of external current we normalized τ by the firing period T which depends on I. Both AS and the DS-AS transi- tion can be found in a large region of parameter space. conductance values gI our motif can exhibit different be- haviors as shown in Fig. 2. In order to characterize each Typically, AS occurs for gI > gE. The relation between S τ/T , excitatory and inhibitory conductances for different regime, we define ti as the time in which the membrane potential of the Sender neuron is larger than the thresh- values of I is shown in Fig. 4. Results are independent of initial conditions and perturbations. The phase-locking old (vi ≥ 30 mV) in the i-th cycle (i.e. its i-th spike R regimes are always reached after a transient time. time), and ti as the spike time of the Receiver neuron M It is worth to mention that the presence of the autapse which is nearest to ti . The spike-timing difference τ between S and R is defined as: in the post-synaptic neuron, even for very small values of inhibitory conductance, decreases the spike-timing dif- R S τi ≡ ti − ti . (8) ference between the neurons. This means that autapses could be biologically important to overcome synaptic de- When τi converges to a constant value τ, a phase- lays between distant areas. Moreover, for enough inhi- locked regime is reached [39]. By definition, if τ > 0 bition, the neurons can fire exactly at the same time. we say that the system exhibits delayed synchronization Therefore, autapse can be also considered as a mecha- (DS) and the neurons fire in the usual pre-post order nism to promote zero-lag synchronization[40, 41] which (see upper panels in Fig. 2a and b, for gI =0.15 nS) . If does not require bidirectional connections. τ < 0 we say that anticipated synchronization (AS) oc- We show here that autaptic weights may control the curs and the neurons fire in a non-intuitive post-pre order spike-timing difference between pre and post-synaptic 4 neurons through a conductance induced DS-AS tran- τi. In fact, for this region of the parameter space the sition. This result can be computationally considered inhibition can silence the neuron. as the other way around of the spike-timing dependent The effect of the autapse in the internal dynamics of plasticity (STDP) [42], which is a biological mechanism the Receiver can also be studied for the uncoupled case that allows the spike-timing differences between pre and gE = 0. For this situation, as we increase gI , the firing post-synaptic neurons to control changes in the synaptic period of the neuron subjected to the autapse initially weights. decreases and T/T0 < 1 for any value of external current I (see Fig. 6). This result is in agreement with previous studies showing that the mechanism underlying antici- 3 0.4 pated synchronization is a faster internal dynamics of (a) I = 5 pA (b) I = 6 pA (c) I = 7 pA 2 0.2 the Receiver [19, 32, 43]. \T
(nS) 0
τ The inhibitory autapse can also promote the transition E g 1 -0.2 from DS to AS when the sender and the receiver neu- -0.4 rons are not described by the same equations or param- 3 eters. If the receiver neuron is a fast spiking neuron [35] 0.4 (d) I = 8 pA (e) I = 9 pA (f) I = 10 pA (with model parameters a = 0.01, b = 0.2, c = −65 and 2 0.2 d = 2) the transition from positive to negative τ is still \T
(nS) 0 τ E
g 1 -0.2 continuous and smooth. In such case the external cur- -0.4 rent in each neuron should be different in order to ensure 0 1 2 3 0 1 2 3 0 1 2 3 that their natural frequencies are similar. For example, if g (nS) g (nS) g (nS) −1 I I I IS = 16 pA, IR =4.5 pA, gE =0.3 nS, βI =0.188 ms , −1 βE = 0.3 ms the system exhibits DS if gI . 0.1 and AS if 0.1 . gI . 0.5 nS (data not shown). Moreover, if the receiver neuron is modeled as a two- FIG. 4: (Color online) Normalized spike-timing dif- compartment neuron, the system can also exhibit AS. We ference τ/T (right bar) in the (gI ,gE) projection of have considered that the compartments are connected parameter space for different values of external current I: by an electrical synpase with synaptic current given by DS (blue, mostly at the upper part in which gI > gE and τ/T < 0), AS (red, middle), and PD (white, meaning that no I = gelectrical ∗ (Vpre − Vpost) and the second compart- stationary value of τ was found). ment sends an inhibitory chemical synapse to the first. For gE = 1 nS, gelectrical =0.2 nS, I = 10 pA the system undergoes a transition from DS to AS when we increase gI from zero to 2.5 nS (data not shown). Further in- B. GABAergic autapse promotes a faster internal vestigation on how anticipated synchronization depends dynamics on more complex dendritic trees, other types of neurons and slower synaptic currents mediated by NMDA and GABA are beyond the scope of this paper and should Occasionally inhibitory synapses are viewed as a bio- B be reported elsewhere in the future. logical mechanism to avoid neurons to fire. However, we show here that the GABAergic autapse can allow the neuron to fire with higher frequencies. In the phase- locking regimes (both DS and AS) the period of the Re- IV. CONCLUDING REMARKS ceiver is the same of the Sender (TR = TR = T0 which is the period of a neuron without an autapse gI = 0). As To summarize, we have shown that the presence of we increase the inhibition, the spike-timing difference de- a GABAergic autapse can promote anticipated synchro- creases and eventually, the system loses the phase-locking nization between two coupled neurons. Here, the in- regime. For gE = 0.3 nS and I ≥ 8 pA, as we increase hibitory self-innervation acts as a dynamical delayed neg- gI the system exhibits a second transition from AS to a ative self-feedback [13, 23] or the inhibitory loop medi- phase-drift regime in which TR
(a) perception [51, 52]. Differently from the first papers 1.005 about AS [13, 23], here the anticipation time is not hard- wired in the dynamical equations, but rather emerges from the autapse dynamics. The spike-timing difference 1.000 between post- and pre-synaptic neurons decrease as we
0 increase the inhibitory autaptic conductance. Eventu-
/T 0.995
R ally the system presents a conductance induced transition T from delayed to anticipated regimes similarly to what has 0.990 been reported in a three neuron motif [25, 28]. This AS- DS transition could synergistically work together with spike-timing-dependent plasticity [27, 42, 53] to deter- 0.985 1.000 0 1 2 3 gI (nS) (b) 0.996 0
IMS = 5 pA T/T IMS = 6 pA 0.992 IMS = 7 pA IMS = 8 pA IMS = 9 pA IMS =10 pA 0.988 0 1 2 3 gI (nS)
FIG. 6: (Color online) Inhibitory autapse promotes a faster internal dynamical of the free-running Re- ceiver, which could be the mechanism promoting AS and FIG. 5: (Color online) Characterizing the phase-drift the transition from DS to AS. For the uncoupled situation regime induced by the inhibitory autapse: the Re- (gE = 0) the period of the free-running Receiver (normal- ceiver is faster than the Sender. (a) The mean period ized by its own period for gI 0, TR/T0) decreases as we in- of the Receiver normalized by its own period when gI = 0, crease the inhibitory conductance gI . For external currents TR/T0 coincides with the mean period of the Sender for DS I = 5, 6, 7 pA the minimum value of T/To can also be related and AS regimes, but it is smaller for PD. (b) In PD, the return to the minimum value of τ in Fig. 3. For I ≥ 8 pA the system map of the period in each cycle of the Receiver is consistent undergoes a transition from AS to PD regime before TR/T0 with a quasi-periodic system. reaches the minima.
AS transition can be a faster internal dynamics of the mine the circuit dynamics and synaptic conductances. Receiver system. Moreover, previous work on AS have Including effects from synaptic plasticity in the model investigate the phenomenon for neurons with firing rates presented here is a natural next step which we are cur- larger than 60 Hz [25, 27]. Typically cortical neurons rently pursuing. in vivo do not fire at rates greater than ∼ 30 Hz. Here we show that AS can occur for firing rates around 20 Hz which means that AS is a robust phenomenon at different Acknowledgments time scales. The DS-AS transition could possibly explain com- The authors thank FAPEAL and UFAL (for PIBIC monly reported short latency in visual systems [44–49], grant), CNPq (grant 432429/2016-6) and CAPES (grant olfactory circuits [50], songbirds brain [34] and human 88881.120309/2016-01) for finnancial support.
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