Chaotic dynamics in neural systems Krishna Pusuluri, Huiwen Ju and Andrey Shilnikov Abstract Several basic mechanisms of chaotic dynamics in phenomenological and biologically plausible models of individual

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Chaotic dynamics in neural systems Krishna Pusuluri, Huiwen Ju and Andrey Shilnikov Abstract Several basic mechanisms of chaotic dynamics in phenomenological and biologically plausible models of individual neurons are discussed. As parameters of a model are varied, chaos occurs at the transition boundaries between generic activity types such as tonic spiking, bursting and quiescence, and the model may also become bi-stable with coexisting behaviours at such transitions. The bifurcations underlying these basic transitions can give rise to period-doubling cascades, various homoclinic and saddle phenomena, torus-breakdown, and chaotic mixed mode oscillations in neuronal systems. 1 Introduction Neurons exhibit various activity regimes and state transitions that reflect their intrinsic ionic channel behaviors and modulatory states. The fundamental types of neuronal activity can be broadly defined as quiescence, subthreshold and tonic spiking oscillations, as well as bursting composed of alternating periods of spiking activity and quiescence. A single neuron can endogenously demonstrate various bursting patterns, varying in response to the external influence of synapses, or to the intrinsic factors such as channel noise. The co-existence of bursting and tonic spik- Krishna Pusuluri Neuroscience Institue, Georgia State University, Petit Science Center, 100 Piedmont Avenue, Atlanta, Georgia 30303, USA. Huiwen Ju Neuroscience Institue, Georgia State University, Petit Science Center, 100 Piedmont Avenue, Atlanta, Georgia 30303, USA. Andrey Shilnikov Neuroscience Institute, and Department of Mathematics and Statistics, Georgia State University, Petit Science Center, 100 Piedmont Avenue, Atlanta, Georgia 30303, USA. 1 2 Krishna Pusuluri, Huiwen Ju and Andrey Shilnikov ing, as well as several different bursting modes have been observed in modeling [6, 13, 14, 15, 16] and experimental [17, 18, 19] studies. This complexity enhances the flexibility of the nervous and locomotive systems [20]. The functional role of chaotic behaviors, and the dynamical and bifurcational mechanisms underlying their onset at transitions between neural activity types like spiking, bursting and quiescence, have been the focus of various theoretical and experimental studies. Bursting is a manifestation of multiple time scale dynamics, composed of repetitive fast tonic spiking and a slow quiescent phase. It has been observed in various fields of science as diverse as food chain ecosystems [2], nonlinear optics [3], medical studies of the human immune system [4], and neuroscience [5, 6]. Various bursting patterns, whether regular or chaotic, endogenous or as emergent network phenomena, are the natural rhythms generated by central pattern generators (CPG) [94, 103]. CPGs are neural networks made up of a small number of constituent neurons that often control various vital repetitive locomotive functions [8] such as walking and respiration of humans, or the swimming and crawling of leeches [94]. Polyrhythmic bursting dynamics have also been observed in multifunctional CPG circuits that produce several coexisting stable oscillatory patterns or bursting rhythms, each of which is associated with a particular type of locomotor activity of the animal [124, 123]. Bursting has also been frequently observed in pathological brain states [9, 10], in particular, during epileptic seizures [11, 12]. Neurons in bursting modes differ in their ability to transmit information and respond to stimulation from those in tonic spiking mode and therefore play an important role in information transfer and processing in normal states of the nervous system. Understanding and modeling the generic mechanisms regulating the neuronal connectivity and the transitions between different patterns of neural activity, including global bifurcations occurring in neuron models and networks, pose fundamental challenges for mathematical neuroscience, with a number of open problems [25]. The range of bifurcation and dynamical phenomena underlying bursting transcends the existing state of the theory [13, 41, 45, 49, 50, 51, 52, 53, 54, 55, 56, 57]: this includes the blue sky catastrophe, torus-canard formation and breakdown, and homoclinic inclination/orbit-flip bifurcations, all of which can occur on the transition route to bursting in most square-wave and elliptic bursters. Studies of bursting require nonlocal homoclinic bifurcation analysis, which is often based on the Poincaré return mappings [58]. Return mappings have been employed for computational neuroscience in [51, 52, 60, 61]. A drawback of mappings constructed from time series is their sparseness, as they reflect only the dominating attractors of a system. In some cases, feasible reductions to one or two dimensional mappings can be achieved through slow fast scale decomposition of the phase variables for the system [58, 59, 62]. A new, computer assisted method for constructing a complete family of onto mappings for membrane potentials, for a better understanding of simple and complex dynamics in neuronal models, both phenomenological and of the Hodgkin Huxley type, was proposed in [55, 63]. With this approach one can study, for example, the spike- adding transitions in the leech heart interneuron model, and how chaotic dynamics in between is associated with homoclinic tangle bifurcations of some threshold saddle periodic orbits [55]. Qualitative changes in a system s ac- Chaotic dynamics in neural systems 3 tivity at transitions often reveal the quantitative information about changes of certain biophysical characteristics associated with the transition. This approach has proven to be exemplary in neuroscience for understanding the transitions between silence and tonic-spiking activities [22]. Moreover, knowledge about the bifurcation (transition) predicts cooperative behavior of interconnected neurons of the identified types [65]. In this chapter, we discuss nonlocal bifurcations in generic, representative models of neurodynamics, described by high order differential equations derived through the Hodgkin-Huxley formalism. We consider a number of neuroscience-related applications to reveal a multiplicity of causes and their bifurcation mechanisms leading to the onset of complex dynamics and chaos in these models. 2 Neuronal activities and transition mechanisms This chapter deals with neuronal models, both biologically plausible and phenomenological, that can produce complex and distinct dynamics such as tonic spiking, bursting, quiescence, chaos, and mixed-mode oscillations (MMOs) representing fast spike trains alternating with subthreshold oscillations. MMOs are typical for many excitable systems describing various (electro)chemical reactions, including the famous Belousov-Zhabotinky reaction, and models of elliptic bursters [52, 117]. Geometrical configurations of slow-fast neuron models for bursting were pioneered in [21, 22, 23] and further developed in [24, 25, 26, 27]. Dynamics of such singularly perturbed systems are determined by and centered around the attracting pieces of the slow motion manifolds. These are composed of equilibria and limit cycles of the fast subsystem [28, 29, 30, 31, 32, 33, 34, 35], that in turn constitute the backbones of bursting patterns in a neuronal model. Using the geometric methods based on the slow-fast dissection, where the slowest variable becomes a control parameter, one can detect and follow the branches of equilibria and limit cycles in the fast subsystem. The slow-fast decomposition allows for drastic simplification, letting one clearly describe the dynamics of a singularly perturbed system. A typical Hodgkin-Huxley model possesses a pair of such manifolds [21, 34]: quiescent and tonic spiking, respectively. The slow-fast dissection has been proven effective in low-order mathematical models of bursting neurons far from the bifurcation points. However, this approach does not account for the reciprocal, often complex interactions between the slow and fast dynamics, leading to the emergence of novel dynamical phenomena and bifurcations that can only occur in the whole system. Near such activity transitions, the bursting behavior becomes drastically complex and can exhibit deterministic chaos [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. Slow-fast decomposition Many Hodgkin-Huxley type models can be treated as a generic slow-fast system 4 Krishna Pusuluri, Huiwen Ju and Andrey Shilnikov x = F(x, z) z = µg(x, z, α), (1) where and 0 µ 1, x R n, n 2, and z is a scalar, or can be a vector in R 2 (see the extended Plant model with two slow variavles below); α is a control parameter shifting the slow nullcline, given by G(x, α) = 0, in the phase space. In the singular limit µ = 0, the z-variable becomes a parameter of the fast subsystem to detect and continue the equilibrium state (ES), given by F(x, z) = 0, and the limit cycles (LC) of the fast subsystem. As long as they (ES/LC) remain exponentially stable, by varying z one can trace down the smooth invariant manifolds in the phase space of (1) such as M eq with the distinct Z-shape typical for many Hodgkin-Huxley type models (see Fig. 1), while the limit cycles form a cylinder-shaped surface M lc. Locally, either is a center manifold for (1) persisting in a closed system, in virtue of [28, 29, 30]. The stable upper and lower branches of M eq correspond to the de- and hyperpolarized steady states of the neuron, respectively. Folds on M eq correspond to the saddle-node equilibrium states of the fast subsystem. The unstable de-polarized branch of M eq can be enclosed by the tonic-spiking manifold M lc typically emerging through a Andronov-Hopf bifurcation and terminating through a homoclinic bifurcation, which are the key features of the fast-subsystem of the square-wave bursters [47, 48], like the Hindmarsh-Rose model [56] and the Chay model [60] (discussed below). Poincaré mappings To elaborate on the nature of complex oscillations like bursting and their evolutions, one needs to examine nonlocal bifurcations that often require the use of Poincaré return maps [39, 51, 52, 60, 78]. An obvious drawback of maps constructed from voltage time series is in their sparseness, as they can typically reveal some pointwise attractors of the system that trajectories fast converge to, unless there is a noise or small perturbations are added to get a more complete picture of the underlying structure. In some cases, a feasible reduction to low-dimensional mapping can be achieved through slow fast scale decomposition of slow phase variables [58, 59, 62]. We proposed and developed a new computer assisted algorithm for constructing a dense family of onto mappings for membrane potentials in a Hodgkin Huxley type neuronal model [55]. Such maps let us find and examine both the stable and unstable solutions in detail; moreover, unstable points are often the primary organizing centers globally governing the dynamics of the model in question. The construction of such a map begins with the localization of the tonic spiking manifold M lc in the model, using the parameter continuation technique or the slow-fast dissection, see Fig. 1. Then, a curve on M lc is defined, which corresponds to minimal (maximal) voltage values, denoted, say, by V 0. By construction, the 1D map M takes all V 0 (outgoing solutions integrated numerically) on this curve back onto itself, after a single turn around M lc, i.e., M : V 0, V 1 for a selected value of the parameter. Two such maps are depicted in Fig. 1,2. One can see that these are non-invertible [81, 82], unimodal maps with a single critical point [79, 80], which Chaotic dynamics in neural systems 5 happens to be a universal feature of many other square-wave bursters in neuroscience applications. With such maps, one can fully study the attractors, the repellers and their bifurcations, including saddle-nodes, homoclinic orbits, spike-adding, and period-doubling. We note that detection of homoclinics of a saddle periodic orbit in the phase space of a model is in general state-of-the art and the Poincaré map technique allows us to locate them with ease. Classifications of bursting The existing classifications [21, 22, 23, 24, 26, 27] of bursting are based on the bifurcation mechanisms of dynamical systems in a plane, which initiate or terminate fast trajectory transitions between the slow motion manifolds in the phase space of the slow-fast neuronal model. These classifications allow us to single out the classes of bursting by subdividing mathematical and realistic models into the following subclasses: elliptic or Hopf-fold subclass (FitzHugh-Rinzel [117] and Morris-Lecar models, models of rodent trigeminal interneurons, dorsal root ganglion, and mesencephalic V cells of the brainstem); square-wave bursters or fold-homoclinic subclass (Hindmarsh-Rose model [56, 121], models of pancreatic β-cells, cells in the pre-botzinger complex, as well as intrinsically bursting and chattering neurons in neocortex); parabolic or circle-circle subclass (model of R15 cells in the abdominal ganglion of the mollusk Aplysia, the reduced leech interneuron model at certain parameter values); and fold-fold subclass, or top hat models [36], including the reduced heart interneuron model (3) discussed below. Transition routes The current description of the transition routes between tonic spiking and bursting activities is incomplete and remains a fundamental problem for both neuroscience and the theory of dynamical systems. The first theoretical mechanism revealed in [38], explained chaos in the so-called square wave bursters [21] emerging between tonic-spiking and bursting. Later, two global bifurcations that occur at the loss of stability of a tonic spiking periodic orbit through quite novel homoclinic saddle-node bifurcations, were discovered and explained. The first transition, reversible and continuous, found in the reduced model of the leech heart interneuron [45, 54] and in a modified Hindmarsh-Rose model of a square-wave burster [56, 58], is based on the blue sky catastrophe [58, 66, 67, 68, 69]. This was proven in [59] to be a typical bifurcation for slow-fast systems. This striking term [70], the blue sky catastrophe, stands for a novel bifurcation of a saddle-node periodic orbit with a 2D unstable manifold returning to the orbit making infinitely many revolutions. After the bifurcation, this homoclinic connection transforms into a long bursting periodic orbit with infinitely many spikes. The burst duration of the orbit near the transition is evaluated by 1/ α, where 0 α 1 is a bifurcation parameter. The second transition mechanism is due to a saddle-node periodic orbit with non-central homoclinics [71]. 6 Krishna Pusuluri, Huiwen Ju and Andrey Shilnikov An important feature of this transition is the bi-stability of co-existing tonic spiking and bursting activities in the neuron model, see Fig. 1. In this case, the burst duration towards the transition increases logarithmically fast, as ln(α). Another feature of this bifurcation is the transient chaos where the neuron generates an unpredictable number of burst trains before it starts spiking tonically. This phenomenon is a direct consequence of the Smale horseshoe finite shift dynamics in the system [72], which is a rather atypical phenomenon for such slow-fast systems. Fig. 1 (A1 ) Bistability of the coexisting tonic-spiking and bursting in the 3D phase space of the Leech heart interneuron model (3). A2 depicts the shape of the corresponding 1D Poincaré map with stable fixed point and 7-period orbit, and 2 unstable fixed points (red): left one separates the quiescent and tonic-spiking phase of bursting. The middle one separates the basins of tonic-spiking (A4 ) and bursting (A3 ). This figure is adapted from [119]. 3 Chaos in neuron models In this section, we present the basic mechanisms and routes to chaos in a variety of biophysically realistic neuronal models exhibiting rich and complex dynamics including tonic spiking, bursting and quiescence. A bifurcation describing a transition between neuronal activities typically occurs near saddle (unstable) orbits and results from reciprocal interactions involving the slow and fast dynamics of the model. Such interactions lead to the emergence of new dynamical phenomena and bifurcations that can occur only in the full model, but not in either of the slow or the fast subsystem. Chaotic dynamics can be characterized by unpredictable variations in the number of spikes during the active phases of bursting and/or the subthreshold Chaotic dynamics in neural systems 7 oscillations. This phenomenon of chaotic dynamics is generally atypical in slowfast systems as it occurs within narrow parameter windows only near the transition boundaries. Indeed, robust and regular dynamics of slow fast neuron models contrast those of real bursting neurons exhibiting a phenomenal time dependent variability of oscillatory patterns. Fig. 2 Chaotic bursting in the Leech heart interneuron model (3) and the corresponding map at a transition between four and five spikes per burst due to primary homoclinics (red lines). This figure is adapted from [119]. Leech heart interneuron model: period doubling cascades and the blue sky catastrophe We first illustrate and discuss the onset of chaotic dynamics in the reduced (3D) model of the leech heart interneuron (see equations (3) of Appendix). This is a typical slow-fast Hodgkin-Huxley type (HH) model describing the dynamical interplay of a single slow variable persistent potassium current, IK2, and two fast variable the sodium current, INa and the membrane voltage V that can be recast in this generic form [54, 46, 118, 119]: CVi0 = 3+ X x Ix K X syn Ik, τ x x 0 = x (V ) x, (2) k This model shows a rich set of dynamics and can produce various types of complex chaotic and bistable behaviors, including the period-doubling cascade enroute tonic spiking through bursting [45, 48], as well as various types of homoclinic chaos. Following the period-doubling cascade, the model demonstrates a terminal phase of chaotic tonic spiking that coexists alongside another periodic tonic spiking activity. 8 Krishna Pusuluri, Huiwen Ju and Andrey Shilnikov Fig. 3 Bi-parameteric sweep of the Leech heart interneuron model (3) using the symbolic toolkit Deterministic Chaos Prospector to process wave-form traces [127, 126, 125] reveals regions of quiescent behavior, tonic spiking, as well as bursting with spike adding cascades. The noisy regions near the boundaries of spike addition reveal the occurrence of chaos, while the noisy boundary between tonic spiking and bursting portrays the blue sky catastrophe. For a different set of parameter values compared to the period doubling cascade, the model can also exhibit the blue sky catastophe as a continuous and reversible mechanism of the transition between bursting and tonic spiking (). Fig. 1 explains the nature of bi-stability in this model as it exhibits the co-existing tonic-spiking and bursting oscillations corresponding to the stable fixed point (FP) (purple) and the period-7 orbit in the 1D map, whose basins are separated by an unstable FP representing a saddle periodic orbit (red) on the 2D manifold M lc in the 3D phase space. The role of the other unstable (red) FP is revealed by Fig. 2. It is shown that the spike-adding in bursting is accompanied with an onset of chaotic dynamics orchestrated by the homoclinic orbits and bifurcations involving the other saddle orbit, see more details in [116, 117, 118, 119, 120, 121, 122]. Fig. 3 shows the bifurcation diagram of the system constructed as a parameteric sweep using our previously developed symbolic toolkit called the Deterministic Chaos Prospector to process symbolic sequences extracted from wave-form traces [127, 126, 125]. This bifurcation diagram identifies the regions of quiescence, tonic spiking, as well as bursting with spike adding cascades. The noisy regi

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