# User:Andrey Shilnikov/Proposed/Multi-stability in neuronal models

In preparation

The ability of distinct anatomical circuits to generate multiple patterns of rhythmic activity is widespread among vertebrate and invertebrate species. These patterns correspond to different locomotor behaviours. For example, swimming and struggling behaviour of the tadpole are controlled by two very different activity patterns of the spinal locomotor generator. In computational neuroscience, such multifunctional neuronal networks are modelled by multistable activity dynamics and modular organization of neural circuits. The neurobiological evidence suggests the multifunctional circuits can be classified by distinct architectures, yet the activity patterns of individual neurons involved in more than one behaviour can vary drastically. In computational neuroscience, such multifunctional neuronal networks are modelled by multi-state dynamics and modular organization of neural circuits. Each state corresponds to an attractor co-existing in the phase space of the model describing the neuronal activity; transitions between these states triggered by external perturbations are represented by switching between the corresponding attraction basins. The neurobiological evidence suggests multifunctional circuits can be classified by distinct architectures, yet the activity patterns of individual neurons involved in more than one behaviour can vary drastically. Several mechanisms, including external (e.g. sensory) input, the activity of projection neurons, neuromodulation, and biomechanics, are responsible for the switching between patterns. This complexity of neuronal dynamics adds potential flexibility to the nervous system; and multi-stability has many implications for motor control. Recent advances in both experimental techniques and modelling tools form a basis for studying these complex circuits. Understanding generic mechanisms of evolution of neuronal connectivity and transitions between different patterns of neural activity is a fundamental problem of neurobiology.

Figure 1: Tonic spiking and bursting in the phase space of the reduced leech interneuron model.

Multistability remains an intriguing phenomenon for neuroscience. The functional roles of multistability in neural systems are not well understood. On the other hand, its pathological roles are widely discussed in recent studies. Knowledge of the dynamic mechanisms supporting multistability opens new horizons for medical applications. It has been intensively targeted in a search for new treatments of the medical conditions which are caused by malfunction of the dynamics of the CNS. Sudden infant death syndrome, epilepsy and Parkinson’s disease are examples of such conditions. Recent progress in the modern technology of computer-brain interface based on real time systems allows to utilize complex feedback stimulation algorithms suppressing pathological regime co-existing with the normal one.

It still remains a challenge to identify the scenarios leading to the multistability in the neuronal dynamics and discuss what are potential roles of the multistability in the operation of the central nervous system under normal and pathological conditions. Multistability of neuronal regimes of activity has been studied combining theoretical and experimental approaches since the pioneering works by Rinzel, 1978 and Guttman et al. 1980. Multistability means the co-existence of attractors separated in the phase space describing the state of neuronal system. The multistability is actively studied on both levels, cellular and network. On the cellular level, it is co-existence of different basic regimes like bursting, spiking, sub-threshold oscillations and silence. On the network level, examples of the multistability include co-existence of different synchronization modes, “on” and “off” states, and polyrhythmic oscillations. Generically, one can design stimulation procedures which will switch system between regimes. Also, multistability suggests presence of hysteresis in response to neuromodulation.

The complexity of neuronal dynamics originates from dynamical diversity of ionic and synaptic currents, which can be separated by different time scales; and multistability of neuronal dynamics can be described within a framework of the analytical and computational methods of qualitative theory of slow-fast systems and the theory of bifurcation. It remains a challenge for the interdisciplinary scientific community to identify all possible bifurcations giving rise to bursting and other regimes of complex dynamics and to describe all possible scenarios supporting co-existence of different regimes of activity

The functional roles of multistability in neural systems are not well understood. On the other hand, its pathological roles are widely discussed in recent studies. Knowledge of the dynamic mechanisms supporting multistability opens new horizons for medical applications. It has been intensively targeted in a search for new treatments of the medical conditions which are caused by malfunction of the dynamics of the CNS. Sudden infant death syndrome, epilepsy and Parkinson’s disease are examples of such conditions. Recent progress in the modern technology of computer-brain interface based on real time systems allows to utilize complex feedback stimulation algorithms suppressing pathological regime co-existing with the normal one.

## Types of multistability

Figure 2: Bifurcation diagram of the reduced model of a leech heart interneuron showing three zones of bistability: tonic spiking and bursting, spiking and hyperpolarized quiescence, depolarized quiescence and bursting

Neurons are observed in one of four fundamental, generally defined modes: silence, sub-threshold oscillations, spiking and bursting. The co-existence of these modes has been observed in modeling and experimental studies of single neurons and neuronal networks. Such multi-stability may be controlled by neuromodulators and thus reflect the functional state of the nervous system. This complexity adds potential flexibility to the nervous system; and multi-stability has many implications for motor control, dynamical memory, information processing, decision making and pathological conditions in central nervous system.

## Tonic spiking and Tonic Spiking

Figure 3: Coexistent tonic spiking attractors in the reduced model of a leech heart interneuron.

## Tonic spiking and Bursting

Figure 4: Coexistent tonic spiking attractors in the reduced model of a leech heart interneuron.
Figure 5: Coexistent tonic spiking attractors in the reduced model of a leech heart interneuron.

## Reduced leech heart interneuron model

Reduced oscillatory heart interneuron model [A. Shilnikov and Cymbalyuk, 2005]: $\tag{1} \mathrm{\dot V} = \mathrm{-2\,[30\, m^2_{K2} (V+0.07)+8\,(V+0.046)}+ \mathrm{200\, f^3_{\infty}(-150,\,0.0305\,,V) h_{Na}\,(V-0.045)}+0.0060]\ ,$

$$\mathrm{\dot h_{Na}} = \mathrm{[f_{\infty}(500,\,0.0325,\,V)-h_{Na}]/0.0406}\ ,$$


$\mathrm{\dot m_{K2}} =\mathrm{[f_{\infty}(-83,V_{\frac{1}{2}}+V_{K2}^{shift},V)-m_{K2}]/0.9}\ ,$ where $$\mathrm{V}$$ is the membrane potential, $$\mathrm{h}_{\rm Na}$$ is inactivation of the fast sodium current, and $$\mathrm{m}_{\rm K2}$$ is activation of persistent potassium one; a Boltzmann function $$\mathrm{f_{\infty}(a,b,V)=1/(1+e^{a(b+V)})}$$ describes kinetics of (in)activation of the currents. The bifurcation parameter $$\mathrm{V^{shift}_{K2}}$$ is a deviation from the canonical value $$\mathrm{V_{\frac{1}{2}}}=0.018$$V corresponding to $$f_{\infty}=1/2\ ,$$ i.e. to the semi-activated potassium channel.

## Polyrhythmicity

Figure 6: Motif as an element of a Cenral Pattern Generator comprised of bursting interneuron
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