Large period spiking and bursting in an excitable system with memory

. Excitability in dynamical systems refers to the ability to transition from a resting stationary state to a spiking state when a parameter is varied. It is the mechanism behind spike generation in neurons. Optical non-linear resonators can be excitable systems, but they usually present a fast response compared to neuronal systems, and they prove di ffi cult to observe experimentally. We propose investigating optical resonators with delayed Kerr e ff ects, specifically in two di ff erent geometries: an oil-filled single-mode cavity with thermo-optical nonlinearity, and two coupled, symmetrically driven cavities. When the Kerr e ff ect is delayed, even a single cavity exhibits excitability. However, we show that it su ff ers from limitations on the thermo-optical relaxation time in order to be realized experimentally. We overcome these limitations using the geometry with coupled cavities, where the thermo-optical relaxation time acts as a memory. This slow variable enables to tailor the spiking frequency and it mimics neuronal behaviours by enabling large-period spiking.


Introduction
When considering dynamical systems, excitability refers to the ability to transition from a resting stationary state to a spiking state when a parameter is varied.Spiking states are related to limit cycles, a particular type of solution in non-linear dynamics associated with self-sustained oscillations, i.e., periodic responses while the excitation is constant.Excitability is observed in various physical systems, especially in neuroscience, where it underlies the generation of action potentials in neurons.
Excitability in photonics systems has already been observed.However, the periodicity, i.e., the timescale of the spiking behavior, is relatively fast, making it difficult to observe experimentally.Moreover, this fast response does not correctly model the slow dynamics occurring in neurons or how neurons can adapt their spiking frequency.
We propose investigating optical resonators with delayed Kerr effects as excitable systems.Such systems have already been realized experimentally but not in the context of excitability.In particular, we study two different geometries.The first consists of an oil-filled single-mode cavity with thermo-optical nonlinearity [inset of Fig. 1(a)].The second is instead made up of two coupled, symmetrically driven cavities.

Optical resonator based excitable systems
When considering a Kerr cavity, it exhibits bistability when driven in the nonlinear regime [Fig.1(b)].This means that the system can support multiple stable states for the same driving amplitude, leading to possible hysteresis when the detuning or input power is varied.However, when the nonlinearity is instantaneous, i.e., fast compared to the system's dissipation channels, it does not support limit cycles.Therefore, a single cavity with an instantaneous Kerr effect is not excitable, as it does not exhibit self-sustained oscillations when driven with a monochromatic pump at constant input power.This is no longer the case when the Kerr effect is delayed, using, e.g., a thermo-optical nonlinearity with a relaxation time τ.When defining the equations modeling ⋆ e-mail: Bertrand.Braeckeveldt@umons.ac.be such a system using a coupled-mode framework, this relaxation time can be treated using an integro-differential equation where the integral involves a memory kernel as a function of τ.Therefore, τ will be referred to as the system memory time.It has been shown that using a memory time commensurate with the total dissipation rate Γ, the system presents limit cycles [1].So for a constant pump intensity the system can enter a periodic regime exhibiting self-sustained oscillations.As this regime does not appear for all detunings between the pump and the cavity resonance, or for all input powers, the modulation of one of these parameters can be used to exhibit the excitable behavior of the system.
For the single cavity geometry, excitability is limited by the constraint Γτ ≃ 1.As the memory time τ increases compared to the dissipation rate Γ, higher powers and larger detunings are required to observe limit cycles.Therefore, it is impossible to realize excitable systems with large memory using this geometry.
To overcome this major limitation, we propose a new geometry based on coupled cavities.Coupled Kerr cavities have already been widely studied in the literature [2,3], but mainly with the instantaneous non-linearity.Recently it has been shown that coupled cavities present excitability, meaning that isolated spikes can be triggered thanks to the addition of noise in the pump [4].However, the timescale of the spike generation was determined by the system dissipation rate and is therefore quite fast compared to neuronal systems and still difficult to observe experimentally.For the first time, we propose to account for a possible delay in the non-linear response to see how it affects the excitable behaviour of the coupled system and how it can potentially overcome the limitations of the single cavity, i.e., the limited range of memory time that can be considered.The coupling has the effect of separating the cavity resonance into two frequencies, separated by two times the coupling rate J, similar to energy splitting in molecular systems with bonding and anti-bonding states.We propose to drive the system symmetrically, i.e., with the same amplitude and phase for both cavities.Interestingly, despite the symmetric nature of the system, it supports asymmetric solutions for which the light field in the first cavity differs from the one in the second cavity.We show in Fig. 2 that these asymmetric solutions support limit cycles even for large memory times (Γτ ≫ 1), and even at small input power and detuning.Therefore, the coupling enables us to overcome the limitation of the single cavity system, i.e., Γτ ≃ 1.Interestingly, this remains true even for small coupling values e.g., J = 0.5Γ which is not the case for instantaneous Kerr effect where a larger coupling is necessary for excitability [2,4].Moreover, we found that the period of self-sustained oscillation is proportional to the memory time [Fig.2(d)] meaning that we provide a way to tailor the spiking frequency.
The coupled system is a highly energy-efficient excitable system as it exhibits self-pulsing at small powers.We show how the addition of a slow variable like the memory controls the spiking frequency and mimics neuronal behaviours.Indeed, slow firing can be achieved using a large memory time non-linear medium, already presented in the literature [5].Moreover, a modification of the memory time will lead to an adaptation of the spiking frequency, like in neurons [6].Finally, when noise is added to the system, we show that the coupled system exhibits both class I excitability associated with single isolated spikes [Fig.3(a)] and class II excitability related to spike trains [Fig.3(b)], known as bursting [7,8].

Conclusion
Excitable systems based on optical resonators suffer limitations and fail to correctly mimic neuronal spiking.Therefore, we propose a geometry based on coupled cavities with delayed Kerr non-linearity to overcome these limitations.This system supports large period self-sustained oscillations that can be tailored at wish via the system memory time.Such a system could be realized experimentally using an oil-filled cavity presenting large thermooptical nonlinearity (e.g., Γτ ∼ 10 6 ) paving the way for energy-efficient, slow and adaptable neuron-like firing in optical systems.

Figure 1 .
Figure 1.Light field intensity n in a single cavity with Kerr non-linearity versus detuning ∆/Γ.(a) Linear regime.(b) Nonlinear regime with bistable regime in shaded green.Inset shows a schematic oil-filled cavity.

Figure 2 .
Figure 2. (a)-(c) Photon population in cavities 1 and 2 (respectively in red and blue) in a self-sustained oscillation regime for multiple relaxation times τ.(d) Evolution of oscillations period versus relaxation time using a continuation technique.

Figure 3 .
Figure 3. Spiking (a) and bursting (b) behaviours when noise is added to the coupled cavity system with memory.