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ScienceDirect Procedia IUTAM 22 (2017) 160 – 167
IUTAM Symposium on Nonlinear and Delayed Dynamics of Mechatronic Systems
Identifying time delay-induced multiple synchronous behaviours in inhibitory coupled bursting neurons with nonlinear dynamics of single neuron Zhiguo Zhaoa , Huaguang Gua,∗ a School
of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
Abstract Time delay-induced multiple synchronous behaviors are simulated in inhibitory coupled bursting neurons when time delay is larger than the period of the bursting, which can be well interpreted with dynamics of single neuron combined with inhibitory coupling current. The bursting pattern of uncoupled neuron can change to multiple different firing patterns when receiving negative stimulus at suitable phase to form the multiple patterns of synchronous bursting appearing at different time delay. The synchronization is dependent on the stable characteristics of the quiescent state corresponding to the stable node of the fast subsystem, wherein long lasting inhibitory coupling current is conductive to achieve synchronization. This is the reason that inhibitory coupling current modulated by time delay can induce multiple synchronous behaviors. c 2017 2017The TheAuthors. Authors. Published Elsevier © Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the IUTAM Symposium on Nonlinear and Delayed Dynamics of Peer-review under responsibility of organizing committee of the IUTAM Symposium on Nonlinear and Delayed Dynamics of Mechatronic Systems.
Mechatronic Systems
Keywords: Synchronization; Inhibitory coupling; Bursting; Bifurcation; Time delay
1. Introduction Synchronization is ubiquitous in nature, and studied in many fields such as mechanics, physics and biology 1 . In neuroscience, synchronization appeared in many brain regions including hippocampus, thalamus and neocortex 2,3 , and is related to many functions such as learning, memory and locomotion 3 . The synchronous behaviors of the nervous system are dependent on both coupling of the synapse and intrinsic cellular properties of the neuron 2,4 . Inhibitory coupling plays important roles in the central pattern generation to generate motor rhythms or in the central nervous system to maintain the balance between excitatory and inhibitory couplings. It was believed that the reciprocal inhibitory coupling always induces anti-phase synchronization in some previous investigations 5,6 . Some theoretical and biological experimental studies found that slow inhibitory synapse can induce synchronization 6–9 . Time delay is inherent in the nervous system due to the finite propagation speed of information and time lapses in synapse 10 . Time delay can induce multiple synchronizations when time delay is within multiple of the intrinsic period ∗
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2210-9838 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the IUTAM Symposium on Nonlinear and Delayed Dynamics of Mechatronic Systems doi:10.1016/j.piutam.2017.08.021
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of individual neuron 11 or within one period of individual neuron 12,13 . Despite these investigations, the dynamics of inhibitory coupled neurons remained unclear, for example, how to identify the spatiotemporal behaviors of network with the dynamics of single neuron combined with the inhibitory coupling current. Two or three reciprocally inhibitory coupled neurons, for example, the pyloric network of lobster 8 , can generate motor rhythms related to digestive function and can exhibit in-phase bursting synchronization as time constant of synapse becomes long. The bursting pattern of an isolated neuron is resulted from the competition between fast and slow currents and exhibits alternation between fast spikes and slow quiescent state. The complex dynamics of bursting pattern can be acquired with fast/slow variable dissection method 4,14,15 . In the inhibitory coupled bursting neurons, as time delay becomes long, a slow time scale is introduced and will interact with the time scales of the bursting pattern. In the present paper, time delay-induced multiple synchronous behaviours are well interpreted with nonlinear dynamics of single neuron and inhibitory coupling current applied at different phase modulated by different time delay. The nonlinear dynamics includes different responses of bursting pattern at different phase stimulated by the negative current and the stable characteristics of the quiescent state of the bursting pattern, which are acquired with fast-slow variable dissection method. The rest of the paper is organized as follows. Section 2 gives the theoretical models and methods. The results are presented in section 3. The conclusion is provided in section 4. 2. Models and methods The reduced leech heart model 14–16 is widely used as a bursting neuron model and is given by: C V˙ = −[gNa f (−150, 0.0305, )hNa (V − E Na ) − gK2 m2K2 (V − E K2 ) + gL (V − E L ) + Iapp ], h˙ Na = [ f (500, 0.0325, V) − hNa ]/τNa , shi f t , V) − mK2 ]/τK2 . m ˙ K2 = [ f (−83, 0.018 + VK2
(1) (2) (3)
Where V is membrane potential; mK2 and hNa are the gating variables describing the activation of potassium current and inactivation of sodium current, respectively; C is the membrane capacitance; gNa , gK2 and gL are the conductance of sodium current, potassium current and leaky current, respectively; E Na , E K2 and E L are the reversal potential of sodium current, potassium current and leaky current, respectively; τK2 and τNa are the time constants of activation of potassium current and inactivation of sodium current, respectively; f (x, y, z) = 1/(1 + exp{x(y + z)}) is a Boltzmann shi f t function describing kinetics of the currents. VK2 is the shift of the membrane potential of half-activation of potassium current from its canonical value. In the present paper, τNa = 0.0405 s, τK2 = 0.9 s, and other parameter values are C = 0.5 nF, gK2 = 30 nS, gNa = 200 nS, gL = 8 nS, E K2 = −0.07 V, E Na = 0.045 V, E L = −0.046 V, Iapp = 0.001 mA shi f t and VK2 = −0.01. The fast threshold modulatory synapse 5 is employed in the present paper and described as follows: I syn = −g s (V pos (t) − E s )Γ(V pre (t − τ)).
(4)
Where g s and E s are the coupling strength and reversal potential of synapse, respectively. To ensure the coupling is inhibitory, E s = −0.0625 V is chosen. V pre and V pos are the presynaptic and postsynaptic potentials, respectively. Γ(x) = 1/(1 + exp(−1000(x − Θ syn ))), where Θ syn is the threshold of synapse and Θ syn = −0.03 V. τ is time delay. The model of three reciprocal coupled neurons by inhibitory synapse is described as follows: C V˙i = −[gNa f (−150, 0.0305, )hi (Vi − E Na ) − gK2 m2i (Vi − E K2 ) + gL (Vi − E L ) + Iapp + Iisyn ], h˙ i = [ f (500, 0.0325, Vi ) − hi ]/τNa , shi f t , Vi ) − mi ]/τK2 . m ˙ i = [ f (−83, 0.018 + VK2 Where Iisyn = −(Vi (t) − E s ) 3j=1( ji) gi j Γ(V j (t − τ)), g12 = g23 = g31 = g1 and g21 = g32 = g13 = g2 . A similar function S i j is used to describe the synchronous degree of the network and is given by:
S ij =
(Vi − V j )2 [Vi2 V 2j ]1/2
(5) (6) (7)
(8)
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Where represents the time average, i, j=1, 2, 3, and i j. If S i j =0, meaning Vi and V j achieve complete synchronization. To investigate the collective behaviors of the three neurons, S max = max(S 12 , S 23 , S 31 ) is adopted. Due to the coexisting behaviors appearing in the inhibitory network, the probability (P) of S max < 10−4 for 100 trials with different initial values are calculated. The higher the P value is, the easier it is for the network to become synchronization. The Eqs. (1-2) and Eq. (3) of the single bursting neuron model is the fast and slow subsystems, respectively, due to τK2 is much larger than τNa . The underlying dynamics of the bursting can be acquired with fast/slow variable dissection method, wherein mK2 is taken as the bifurcation parameter in the fast subsystem. The bifurcations are calculated with the software XPPAUT. 3. Results 3.1. Bursting pattern of individual neuron and fast/slow variable dissection The fast/slow variable dissection of period-6 bursting (dotted line, clockwise) with period T = 2.894 of individual neuron is shown in Fig. 1(a). For the fast subsystem, the bifurcation curve of equilibrium exhibits a Z-shape. The upper (UB, dashed line), middle (MB, dashed line), and lower (LB, bold line) branches are unstable equilibrium (mK2 < 0.6682), saddle, and a stable node (mK2 > 0.1115), respectively. The intersection point between MB and LB is saddle-node bifurcation point (SN, mK2 ≈ 0.1115). Except the stable node, there is a stable limit cycle (LC, solid circle), which contact MB at mK2 ≈ 0.352 to form a saddle homoclinic bifurcation (SH). The behaviour of the period-6 bursting alternates between the stable limit cycle to form the burst with 6 spikes and the stable node to form the quiescent state. In the present paper, we lay special emphasis on the dynamics of the quiescent state of the bursting, which becomes ”super-stable” when receiving external inhibitory or negative stimulus. We can assume that if the behaviours of two inhibitory coupled neurons are at quiescent state and receive inhibitory coupling current with long duration, the difference between the trajectories of two neurons becomes small, which can easily lead to synchronization. The assumption can be verified with the simulation results in the following subsections. The stable characteristic of the quiescent state corresponding to the stable node of the fast subsystem is the internal cause to achieve synchronization for inhibitory coupled neurons. 3.2. Different bursting patterns induced by inhibitory current applied at different phases If a strong inhibitory current is applied at suitable phase within the period-6 bursting, new firing patterns different from the period-6 bursting can be induced. These suitable phases include the trough after 4th and 5th spikes, correspondingly, period-4 bursting and period-5 bursting can be induced and the trajectory will run across LB. The period-4 bursting is shown by the bold solid line in Fig. 1(a) and (b). Period-5 bursting is not shown here. The phase at the end of the quiescence state (LB) is another suitable phase, and a period-7 bursting can be induced, as shown by the bold solid line in both Fig. 1(c) and (d). It should be noticed that both period-4 bursting and period-7 bursting induced by the negative pulse run along the quiescent state for a relative long time. However, if the inhibitory current which applied at the suitable phase of the period-6 bursting is weak, new firing patterns can not be induced. For example, period-4 bursting can not be induced when the inhibitory current is weak and is applied at the suitable phase (the trough after the 4th spike), as shown in Fig. 1(e) and (f). It shows that the strength of the inhibitory current plays important roles in influencing the bursting patterns. 3.3. Time delay induced-multiple synchronizations In the present paper, the synchronous behaviors are investigated when T < τ < 3T (T =2.894 is the period of the period-6 bursting). When the coupling strength is weak (g1 =0.01 and g2 =0.04), the probability P exhibits high values at two intervals of τ, as shown in Fig. 2(a). Both synchronous behaviours are period-6 bursting, and appears in T < τ < 2T (labeled with B62 ) and 2T < τ < 3T (B63 ), respectively. The bursting pattern when τ= 4.44 (≈ 1.52T ) is shown in Fig. 3(a). When the coupling strength is strong (g1 =0.6 and g2 =0.9), P manifests high values at multiple regions of τ, as shown in Fig. 2(b). Except for the B62 and B63 , other multiple synchronous behaviours appear in both T < τ < 2T
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Fig. 1. Left: fast/slow variable dissection. Right: time series correspond to trajectory of the left and the negative current (dash dot line). (a) Period-6 bursting (dotted line) and burst with 4 spikes (bold solid line). (b) Time series correspond to (a). (c) Period-6 bursting (dotted line) and burst with 7 spikes (bold solid line). (d) Time series correspond to (c). (e) Period-6 bursting (dotted line) and period-6 bursting perturbed by inhibitory current (bold solid line). (f) Time series correspond to (e).
and 2T < τ < 3T . Right to the B62 , for example, when τ ∈ (4.86, 5.70) (T < τ < T ), the synchronized period-(6+7) 6 bursting appears and is labelled with B67 2 , as shown in Fig. 3(b). Right to the B3 , for example, when τ ∈ (7.78, 8.64) (2T < τ < 3T ), the synchronized period-(6+6+7) bursting appears and is labelled with B667 3 , as shown in Fig. 3(c).
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Fig. 2. Time delay-induced multiple synchronizations at different coupling strengths. (a) g1 =0.01 and g2 =0.04. (b) g1 =0.6 and g2 =0.9.
667 Fig. 3. Synchronized bursting patterns (solid line) and coupling current (dotted line). (a) B62 when τ=4.4. (b) B67 2 when τ = 5.6. (c) B3 when when τ=3.68. Other parameters: (a) g =0.01 and g =0.04; (b-d) g =0.6 and g =0.9. τ=8.3. (d) B65 1 2 1 2 2
The phase of the inhibitory coupling current (dotted line) to induce burst with 7 spikes resembles that of the negative current pulse that can induce an individual neuron to generate burst with 7 spikes (Fig. 1(d)), and duration of the inhibitory coupling current is relative long to ensure to achieve synchronization. The label Bnm is used to distinguish different synchronized bursting patterns, where the subscript m = 2 and 3 indicates T < τ < 2T and 2T < τ < 3T , respectively, and the superscript n means the bursting patterns. For example,B67 2 represents the period-(6+7) bursting and B667 the period-(6+6+7) bursting. 3
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5 Fig. 4. Coexisting synchronized bursting patterns (solid line) and coupling current (dotted line).(a) B64 2 when τ=3.46. (b) B2 when τ=3.46. Other parameters: g1 =0.6 and g2 =0.9. Initial values: (a) V1 = 0.0391245, h1 = 0.170871, m1 = 0.562633, V2 = 0.0381903, h2 = 0.13661, m2 = 0.479153, V3 = −0.0487238, h3 = 0.612608, m3 = 0.684606. (b) V1 = 0.0129057, h1 = 0.544159, m1 = 0.723881, V2 = 0.0483696, h2 = 0.504289, m2 = 0.657177, V3 = −0.046159, h3 = 0.931891, m3 = 0.560916.
There are multiple synchronization regions left to B62 . For example, Left to the B62 , τ ∈ (3.62, 3.76), synchronous 65 period-(6+5) bursting (B65 2 ) appears, as shown in Fig. 3(d). Left to the B2 , for example, τ ∈ (3.46, 3.54), synchronous 64 period-(6+4) bursting (B2 ) appears, as shown in Fig. 4(a). The phase of the inhibitory coupling current (dotted line) to induce burst with 4(5) spikes resembles that of the negative current pulse that can induce an individual neuron to generate burst with 4 spikes (Fig. 1(b)), and the inhibitory coupling current also lasts a long time to ensure to achieve synchronization. Similarly, there are multiple synchronization regions left to B63 , the synchronous behaviours of B65 2 and B665 3 can be well understand (detailed bursting patterns not shown here). 3.4. Complex synchronized bursting patterns and coexisting behaviors 667 65 665 64 As can be found from Fig. 2(b), except the B62 , B63 , B67 2 , B3 , B2 , B3 and B2 that can be well interpreted with Fig. 1, there are other complex synchronous bursting patterns, showing that multiple synchronizations are very complex. 5 For example, Left to the B64 2 , τ ∈ (3.34, 3.48), synchronous period-5 bursting (B2 ) appears, as shown in Fig. 4(b). The generation of burst with k (k < 6) spikes is also due to the suppression of inhibitory current applied at the phase after the kth spike, which is the same as that of Fig. 1(a) and (b). 5 There is a overlapping region between region of B64 2 and region of B2 , as shown in Fig. 2(b). In the overlapping 64 5 region, B2 and B2 coexist, as shown in Fig. 4(a) and (b). The inhibitory current have suppression at the quiescent state for a long duration, and enable the network to achieve synchronization. The troughs after 4th and 5th spikes are the two suitable phases.
3.5. Multiple synchronizations in the plane (τ, g2 ) The distributions of P values and multiple synchronous behaviours in the plane (τ, g2 ) for g1 = 0.01 and g1 = 0.6 are shown in Fig. 5(a) and (b), respectively. It shows that the coupling strength plays important roles in influencing time delay-induced multiple synchronizations. More synchronous bursting patterns appear for g1 = 0.6 (Fig. 5(b)) than for g1 = 0.01 (Fig. 5(a)). For example, there are about 10 synchronous bursting patterns appeared in Fig. 5(b), including 667 65 665 655 64 5 7 B62 , B63 , B67 2 , B3 , B2 , B3 , B3 , B2 , B2 , and B3 . However, only 4 synchronous bursting patterns appeared in Fig. 667 5(b), including B62 , B63 , B67 , and B . The P values of some synchronizations in Fig. 5(a) are smaller than those in Fig. 2 3 667 5(b). For example, the P values of both B67 and B 2 3 in Fig. 5(a) are about 0.2, while in Fig. 5(b) are about 1. For both g1 = 0.01 (Fig. 5(a)) and g1 = 0.6 (Fig. 5(b)), with the increase of the coupling strength g2 , multiple synchronization regions and complex synchronous bursting patterns appear, and the P values of synchronization regions with the same patterns increase. For example, no synchronous behaviors appear in Fig. 5(a) when g2 is below 0.01. However, with increasing g2 , the synchronization regions of B62 and B63 appear and the corresponding P values increase. The results indicate that coupling strength g1 and g2 are important for different synchronous behaviours and P values. Indeed, the inhibitory current monotonously increases with the increase of the coupling strengths g1 and g2 .
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Fig. 5. Distribution of P on plane (τ,g2 ) at different g1 . (a) g1 =0.01. (b) g1 =0.6.
So, the generation of the different synchronous bursting patterns is dependent on the inhibitory current. The strong inhibitory current which is conductive to the neuronal network to achieve multiple synchronizations with different bursting patterns. The strong inhibitory current plays two roles. One is that the inhibitory current can induce different bursting patterns, and the other is that long duration of the inhibitory effect within the quiescent state (hyperpolarization) enable the network to achieve multiple synchronizations. If the inhibitory current can not induce new bursting patterns, like Fig. 1(e) and (f), the duration of inhibitory current at the quiescent state is either short or weak, which cannot ensure the network to achieve the synchronization. 4. Conclusion The results present novel phenomenon of time delay-induced multiple synchronous behaviours in inhibitory coupled neurons, which exhibit more complex dynamics than those reported in the previous investigations 11 . The complex synchronous behaviours can be well interpreted with dynamics of single neuron combined with inhibitory coupling current, which can build a relationship between dynamics of single neuron and spatiotemporal behaviours of network. The stable characteristic of the long quiescent state of bursting and different responses of the bursting neuron to the negative stimulus applied at different phase are the most important factors to ensure the achievement of the multiple synchronous behaviours in the inhibitory coupled neurons. The neural circuit which composed of two or three mutually inhibiting neurons exists in neuronal systems such as Central pattern generators (CPGs). CPGs governing various rhythmic activities including cardiac beating and locomotive behaviors via generating different rhythm pat-
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terns 10 . Understanding the emergence of different asynchronous and synchronous rhythms is useful to understand the functions of neuronal circuit. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 11572225 and 11372224. References 1. Arenas A, Diaz-Guilera A, Kurths J, Moreno Y, Zhou CS. Synchronization in complex networks. Phys Rep 2008;469:93-153. 2. Bartos M, Vida I, Jonas P. Synaptic mechanisms of synchronized gamma oscillations in inhibitory interneuron networks. Nat Rev Neurosci 2007;8:45-56. 3. Colgin LL. Rhythms of the hippocampal network. Nat Rev Neurosci 2016;17:239-49. 4. Belykh I, Shilnikov A. When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons. Phys Rev Lett 2008;101:078102. 5. Wang XJ, Rinzel J. Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput 1992;4:84-97. 6. Van Vreeswijk C, Abbott LF, Bard Ermentrout G. When inhibition not excitation synchronizes neural firing. J Comput Neurosci 1994;1:313-21. 7. Wang XJ, Buzsaki G. Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J Neurosci 1996;16:6402-13. 8. Elson RC, Selverston AI, Abarbanel HDI, Rabinovich MI. Inhibitory synchronization of bursting in biological neurons: Dependence on synaptic time constant. J Neurophysiol 2002;88:1166-76. 9. Chow CC. Phase-locking in weakly heterogeneous neuronal networks. Physica D 1998;118:343-70. 10. Struber M, Jonas P, Bartos M. Strength and duration of perisomatic GABAergic inhibition depend on distance between synaptically connected cells. Proc Natl Acad Sci U S A 2015;112:1220-5. 11. Wu YN, Gong YB, Wang Q. Autaptic activity-induced synchronization transitions in Newman-Watts network of Hodgkin-Huxley neurons. Chaos 2015;25:043113. 12. Zhao ZG, Gu HG. The influence of single neuron dynamics and network topology on time delay-induced multiple synchronous behaviors in inhibitory coupled network. Chaos Solitons Fractals 2015;80:96-108. 13. Gu HG, Zhao ZG. Dynamics of time delay-induced multiple synchronous behaviors in inhibitory coupled neurons. PLoS One 2015;10:0138593. 14. Shilnikov A, Calabrese RL, Cymbalyuk G. Mechanism of bistability: Tonic spiking and bursting in a neuron model. Phys Rev E 2005;71: 056214. 15. Shilnikov A, Cymbalyuk G. Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phy Rev Lett 2005;94:048101. 16. Shilnikov A, Gordon R, Belykh I. Polyrhythmic synchronization in bursting networking motifs. Chaos 2008;18:037120.
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