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Phase Bursting Rhythms in Inhibitory Rings Matthew Brooks, Robert Clewley, and Andrey Shilnikov Abstract Leech Heart Interneuron Model 3-Cell Inhibitory Networks References Results and Discussion Further Research Strong Coupling: Symmetric and Asymmetric Motifs Weak Coupling: Symmetric Motifs [1] Shilnikov, A. L., Rene, G., Belykh, I. (2008). Polyrhythmic synchronization in bursting networking motifs. Chaos 18 pp 1-13. [2] Jalil, S., Belykh, I., Shilnikov, A. (2009). Synchronized bursting: the evil twin of the half-center oscillator. PNAS, paper pending. [3] Cymbalyuk, G. S., Calabrese, R. L., and Shilnikov, A. L. (2005). How a neuron model can demonstrate co-existence of tonic spiking and bursting? Neurocomputing 65–66, pp 869–875. [4] Nowotny, T., and Rabinovich, M. I. (2007). Dynamical Origin of Independent Spiking and Bursting Activity in Neural Microcircuits. Phys. Rev. Letters 98, 128107. [5] Ashwin, P., Burylko, O., Maistrenko, Y. (2008). Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Physica D 237, pp 454-466. [6] Cymbalyuk, G., Shilnikov, A. (2005). Coexistence of tonic spiking oscillations in a leech neuron model. J. Comp. Neurosci. 18, pp 255-263. A multifunctional central pattern generator (CPG) is able to produce bursting polyrhythms that determine locomotive activity in an animal: for example, swimming and crawling in a leech. Each rhythm corresponds to a periodic or aperiodic attractor of the CPG. We study the multistability (stable coexistence) of these attractors, as well as the switching between them, using a model of a multifunctional CPG. We consider a Hodgkin-Huxley type model of a leech heart interneuron, three of which are mutually coupled in a ring by fast inhibitory synapses. Each neuron is a 3D system of deterministic ODEs exhibiting periodic bursting, where a burst consists of episodes of fast tonic spiking and slow quiescence. We employ the tools of dynamical systems and bifurcation theory to understand the rhythmic outcomes of the network. We show that the problem can be effectively reduced to the phase plane for the phase differences of the neurons on the bursting periodic orbit. Using computer assisted analysis, we examine the bifurcations of attractors and their basins in the phase plane, separated by repellers and separatrices of saddles which are the hidden organizing centers of the system. These structures determine the resulting bursting rhythms produced globally by the CPG. By varying the coupling synaptic strength, we examine the emerging dynamics and properties synchronization patterns produced by symmetric and asymmetric CPG motifs. C = 0.5 G K2 = 30 E K = -0.07 E Na = 0.045 G Na = 160 G I = 8 E I = -0.046 I pol = 0.006 σ m = 0.0035 σ h = 0.0065 τ K2 = 0.9 τ Na = 0.0405 E syn = -0.0625 Θ syn = -0.03 n = 0.018 h = 0.99 Membrane capacitance, μF K + maximal conductance, nS/μm2 K + reversal potential, V Na + reversal potential, V Na + maximal conductance, nS/μm2 leak maximal conductance, nS/μm2 leak reversal potential, V polarization current, mA K time constant Na time constant inhibitory reversal potential, V Synaptic threshold, V Gating parameter for activation of I K Gating parameter for inactivation of I na The Hodgkin Huxley formulation for the (pharmacologically reduced) leech heart interneuron model is given as: where: V i is the membrane potential, I Na is the sodium current, I K is the potassium current, I leak is the leak current, I pol is the polarization current, I syn is the synaptic current, g ij is synaptic coupling strength between neurons i and j, Γ is the sigmoid coupling function used to drive inhibitory synaptic coupling between neurons For our purposes the excitatory coupling strengths g exc =0 for all neurons in the motif. The motif is strictly driven by inhibitory signals, which are varied in strength. All neurons in the motif are configured identically (see table of parameters). 1 2 3 Asymmetric inhibition: Cells send significantly stronger inhibitory signals in one direction. Each cell in the motifs has a pair of coupling strengths g ij ; i≠j. Due to inhibitory coupling, the motif gives rise to a network period which may differ from the isolated period of a single cell. With respect to neuron 1 (blue) we introduce a pair of phase shifts (Ф 1, Ф 2 ) which reflect the duration of time that cells 2 and 3 are “off”, respectively. The shifts are normalized with respect to the isolated period T iso. Different burst rhythms occur depending on the duration of the phase shifts (Ф 1, Ф 2 ). For instance at (Ф 1, Ф 2 ) = (0.6, 0.3) neuron 1 is anti-phase with respect to neurons 2 and 3. Symmetric inhibition: Cells send inhibitory signals of equal strength in both directions. Φ 1 = 0.6 Φ 2 = 0.3 V t (ms) T iso ≈11.31 T coup ≈ 12.02 t (ms) V Burst rhythm outcomes are computed for discretized values of phase pairs (Ф 1, Ф 2 ) with Ф 1, Ф 2 in [0,1]. Shown above is a symmetric strongly coupled case, with g ij = 0.1, for all i,j. As the phases are varied with respect to T iso, the resulting burst rhythm shifts; one cell in the motif is always anti-phase with the other two. Convergence to each outcome is rapid, often after the first burst cycle completes. + Inhibitory coupling strengths are fixed in the clockwise direction for g ij ={g 21, g 32, g 13 }; g ji ={g 12, g 23, g 31 } are varied identically in increasing magnitude from 0.1 to 0.9. The burst regimes exhibit subtle distortions until g ji ≈ 0.66 (inset C), where a sub-region suddenly appears in the green burst rhythm region, and continues to expand until it becomes tangent to the line Ф 1 = Ф 2 (insets D, E). At g ji = 0.69 (inset F), another region appears, and this process cascades at an increasing rate until g ji =0.705 (inset H), when regions fully desynchronized burst rhythms appear (shown in gray) and the synchronized regions begin to collapse (inset I). By g ij =0.78 desynchronization occurs everywhere for all phase points (Ф 1, Ф 2 ) (voltage trace, inset 4). Ф1Ф1 1: (Ф 1, Ф 2 ) = (0.2, 0.25) 2: (Ф 1, Ф 2 ) = (0.4, 0.9) 3: (Ф 1, Ф 2 ) = (0.8, 0.3) Ф1Ф1 Ф2Ф2 1 2 3 Ф1Ф1 Ф2Ф2 In the case of weakly inhibition, burst rhythms take significantly longer to stabilize, allowing us to see the manner of convergence to the final burst pattern outcome. For weakly coupled motif (g ij =0.0005), there exists well defined regions of both in-phase and desynchronized bursting states. The parameterized phase plot illustrates boundaries where choices of Φ 1 and Φ 2 lead to a specific bursting rhythm, which can be thought of as a stable fixed point in (Φ 1, Φ 2 ). Unstable and saddle activity occurs around the triangular shaped gray regions corresponding to desynchronized burst rhythms. The inset below depicts a voltage trace at (Φ 1, Φ 2 ) = (0.78, 0.31), where desynchronization occurs. t (ms) V Due to symmetric coupling, the regions shown above are symmetric with respect to the line Φ 1 = Φ 2. Since the phase shifts are of unit modulus, the phase shift plot can be thought of as being on a torus (shown right), where convergence to bursting rhythms (i.e. stable fixed points) occurs along the surface. This research is focused on the onset of polyrhythmic dynamics in a model of a multifunctional CPG. Every oscillatory attractor of the network corresponds to a specific rhythm and is conjectured to be associated with a particular type of locomotive activity of a CPG. By elaborating on various configurations of mutually inhibitory and mixed motifs, network building blocks, we intend to describe some universal synergetic mechanisms of emergent synchronous behaviors in CPGs. Each burst rhythm that can be produced by the CPG functions as a oscillatory attractor of the system with respect to the phase shifts of each cell. By varying the strength of the asymmetric coupling in the strongly coupled motif, we observe bursting regimes that ultimately cascade into desynchronized burst rhythms. In order to observe the attractors and repellers of the phase system, we utilize a weak coupling motif that produces a slower rate of synchronization between the burst patterns within the network. Very specific dynamics arise when the phase portrait for the symmetrically coupled case g ij = 0.0005 is computed. There exist 3 stable fixed points corresponding to the known burst rhythm outcomes where one cell is in anti- phase with respect to the others. More notably, there exists a repeller at the origin, which suggests that unless the phase shift is identically (0,0), the burst pattern will always tend to one of the regions (or otherwise be desynchronized). Also of notice is the appearance of unstable focus surrounded by three saddle nodes. Our current hypothesis of the transitioning dynamics is that by shortening the burst (via increasing V K2 shift ) the unstable focus will become stable, by way of all three saddles collapsing onto the focus. We intend to investigate the dynamics that give rise to the cascading burst rhythms for the strongly coupled cases. Additionally, anti-phase (but not necessarily aperiodic) states should yield a series of attractors as well, although these have not been characterized in the work shown. The basins of attraction for the weakly coupled case are significantly different from the strongly coupled case. One possible way to observe this change in dynamics would be to identify the coupling strengths g ij where the system tends from a weakly coupled motif to a strongly coupled one. The evolution of the dynamics of asymmetric weakly coupled motif are not yet known but may yield insight into the bifurcations that give rise to the dynamics described thus far. The measurements made with regard to phase shift are isochronic, i.e. Φ 1 and Φ 2 are discretized with respect to the isolated period. Because of this, more phase shift values are evaluated during the “slow” portion of burst cycle (quiescence) than the “fast” portion (tonic spiking). To rectify this, it has been proposed that the isolated periodic orbit be spliced into equal intervals, from which the phase shift time values would be interpolated. Applying phase shifting to mixed CPG motifs (where inhibitory and excitatory signals are passed) would potentially yield understanding of the complex dynamics generated by those networks. Ф1Ф1 Ф1Ф1 Ф2Ф2 Ф2Ф2 Ф1Ф1 Ф2Ф2 Ф1Ф1 Ф2Ф2 Ф1Ф1 Ф2Ф2 Ф1Ф1 ABC D G E H F I We plot Φ 1 (t), Φ 2 (t), parameterized with respect to time t, indicating the relative phase difference between neuron pairs (blue, green) and (blue, red) respectively. This plot suggests possible mechanics of how the bursting rhythm arrives at the synchronization state. Ф2Ф2 Ф2Ф2 4: (Ф 1, Ф 2 ) = (0.8, 0.5); g ji = 0.8

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