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The Synthetic Nervous System (SNS) is a biologically inspired neural network (NN). Due to its capability of capturing complex mechanisms underlying neural computation, an SNS model is a candidate for building compact and interpretable NN controllers for robots. Previous work on SNSs has focused on applying the model to the control of legged robots and the design of functional subnetworks (FSNs) to realize dynamical systems. However, the FSN approach has previously relied on the analytical solution of the governing equations, which is difficult for designing more complex NN controllers. Incorporating plasticity into SNSs and using learning algorithms to tune the parameters offers a promising solution for systematic design in this situation. In this paper, we theoretically analyze the computational advantages of SNSs compared with other classical artificial neural networks. We then use learning algorithms to develop compact subnetworks for implementing addition, subtraction, division, and multiplication. We also combine the learning-based methodology with a bioinspired architecture to design an interpretable SNS for the pick-and-place control of a simulated gantry system. Finally, we show that the SNS controller is successfully transferred to a real-world robotic platform without further tuning of the parameters, verifying the effectiveness of our approach., Comment: Accepted to ICRA 2023

Living systems can use a single periphery to perform a variety of tasks and adapt to a dynamic environment. This multifunctionality is achieved through the use of neural circuitry that adaptively controls the reconfigurable musculature. Current robotic systems struggle to flexibly adapt to unstructured environments. Through mimicry of the neuromechanical coupling seen in living organisms, robotic systems could potentially achieve greater autonomy. The tractable neuromechanics of the sea slug $\textit{Aplysia californica's}$ feeding apparatus, or buccal mass, make it an ideal candidate for applying neuromechanical principles to the control of a soft robot. In this work, a robotic grasper was designed to mimic specific morphology of the $\textit{Aplysia}$ feeding apparatus. These include the use of soft actuators akin to biological muscle, a deformable grasping surface, and a similar muscular architecture. A previously developed Boolean neural controller was then adapted for the control of this soft robotic system. The robot was capable of qualitatively replicating swallowing behavior by cyclically ingesting a plastic tube. The robot's normalized translational and rotational kinematics of the odontophore followed profiles observed $\textit{in vivo}$ despite morphological differences. This brings $\textit{Aplysia}$-inspired control $\textit{in roboto}$ one step closer to multifunctional neural control schema $\textit{in vivo}$ and $\textit{in silico}$. Future additions may improve SLUGBOT's viability as a neuromechanical research platform., Comment: Submitted and accepted to Living Machines 2022 conference

Motor systems show an overall robustness, but because they are highly nonlinear, understanding how they achieve robustness is difficult. In many rhythmic systems, robustness against perturbations involves response of both the shape and the timing of the trajectory. This makes the study of robustness even more challenging. To understand how a motor system produces robust behaviors in a variable environment, we consider a neuromechanical model of motor patterns in the feeding apparatus of the marine mollusk \textit{Aplysia californica} \citep{shaw2015,lyttle2017}. We established in \citep{WGCT2021} the tools for studying combined shape and timing responses of limit cycle systems under sustained perturbations and here apply them to study robustness of the neuromechanical model against increased mechanical load during swallowing. Interestingly, we discover that nonlinear biomechanical properties confer resilience by immediately increasing resistance to applied loads. In contrast, the effect of changed sensory feedback signal is significantly delayed by the firing rates' hard boundary properties. Our analysis suggests that sensory feedback contributes to robustness in swallowing primarily by shifting the timing of neural activation involved in the power stroke of the motor cycle (retraction). This effect enables the system to generate stronger retractor muscle forces to compensate for the increased load, and hence achieve strong robustness. The approaches that we are applying to understanding a neuromechanical model in \textit{Aplysia}, and the results that we have obtained, are likely to provide insights into the function of other motor systems that encounter changing mechanical loads and hard boundaries, both due to mechanical and neuronal firing properties., Comment: 26 pages, 13 figures

Animals exhibit remarkable feats of behavioral flexibility and multifunctional control that remain challenging for robotic systems. The neural and morphological basis of multifunctionality in animals can provide a source of bio-inspiration for robotic controllers. However, many existing approaches to modeling biological neural networks rely on computationally expensive models and tend to focus solely on the nervous system, often neglecting the biomechanics of the periphery. As a consequence, while these models are excellent tools for neuroscience, they fail to predict functional behavior in real time, which is a critical capability for robotic control. To meet the need for real-time multifunctional control, we have developed a hybrid Boolean model framework capable of modeling neural bursting activity and simple biomechanics at speeds faster than real time. Using this approach, we present a multifunctional model of Aplysia californica feeding that qualitatively reproduces three key feeding behaviors (biting, swallowing, and rejection), demonstrates behavioral switching in response to external sensory cues, and incorporates both known neural connectivity and a simple bioinspired mechanical model of the feeding apparatus. We demonstrate that the model can be used for formulating testable hypotheses and discuss the implications of this approach for robotic control and neuroscience., Comment: Revisions have been made to improve manuscript clarity and expand the introduction and discussion. The results are unchanged

When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface. Examples include heel-ground interaction in locomotion, firing rate rectification in neural networks and stick-slip oscillators. In many rhythmic systems, robustness against external perturbations involves response of both the shape and the timing of the limit cycle trajectory. The existing methods of infinitesimal phase response curve (iPRC) and variational analysis are well established for quantifying changes in timing and shape for smooth systems. These tools have recently been extended to nonsmooth dynamics with transversal crossing boundaries. In this work, we further extend the iPRC method to nonsmooth systems with sliding components, which enables us to make predictions about the synchronization properties of weakly coupled stick-slip oscillators. We observe a new feature of the isochrons in a planar limit cycle with hard sliding boundaries. Moreover, the classical variational analysis neglects timing information and is restricted to instantaneous perturbations. By defining the "infinitesimal shape response curve" (iSRC), we incorporate timing sensitivity of an oscillator to describe the shape response of this oscillator to parametric perturbations. In order to extract timing information, we also develop a "local timing response curve" (lTRC) that measures the timing sensitivity of a limit cycle within any given region. We demonstrate in a specific example that taking into account local timing sensitivity in a nonsmooth system greatly improves the accuracy of the iSRC over global timing analysis given by the iPRC., Comment: 73 pages

The asymptotic phase $\theta$ of an initial point $x$ in the stable manifold of a limit cycle identifies the phase of the point on the limit cycle to which the flow $\phi_t(x)$ converges as $t\to\infty$. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a limit cycle trajectory. For a stable limit cycle in a smooth dynamical system the iPRC is the gradient $\nabla_x(\theta)$ of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the infinitesimal phase response curves (iPRCs) of limit cycles occurring in piecewise smooth (Filipov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, $M$, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth limit cycle systems in which synchronization arises solely due to the crossing of switching manifolds., Comment: 37 pages, 10 figures

Rhythmic behaviors in neural systems often combine features of limit cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small perturbations satisfying a net inflow condition will generically give rise to a stable limit cycle (Reyn, 1980; Guckenheimer and Holmes, 1983). Here we consider the asymptotic behavior of the infinitesimal phase response curve (iPRC) for examples of two systems satisfying Reyn's inflow criterion, (i) a smooth system with a chain of four hyperbolic saddle points and (ii) a piecewise linear system corresponding to local linearization of the smooth system about its saddle points. For system (ii), we obtain exact expressions for the limit cycle and the iPRC as a function of a parameter, mu>0, representing the distance from a heteroclinic bifurcation point. In the limit, as mu approaches zero, we find that perturbations parallel to the unstable eigenvector direction in a piecewise linear region lead to divergent phase response, as previously observed (Brown, Moehlis and Holmes (2004), Neural Computation). In contrast to previous work, we find that perturbations parallel to the stable eigenvector direction can lead to either divergent or convergent phase response, depending on the phase at which the perturbation occurs. In the smooth system (i), we show numerical evidence of qualitatively similar phase specific sensitivity to perturbation., Comment: 32 pages, 10 figures, 57 references

Information theory has explained the organization of many biological phenomena, from the physiology of sensory receptive fields to the variability of certain DNA sequence ensembles. Some scholars have proposed that information should provide the central explanatory principle in biology, in the sense that any behavioral strategy that is optimal for an organism's survival must necessarily involve efficient information processing. We challenge this view by providing a counterexample. We present an analytically tractable model for a particular instance of a perception-action loop: a creature searching for a food source confined to a one-dimensional ring world. The model incorporates the statistical structure of the creature's world, the effects of the creature's actions on that structure, and the creature's strategic decision process. The model takes the form of a Markov process on an infinite dimensional state space. To analyze it we construct an exact coarse graining that reduces the model to a Markov process on a finite number of "information states". This technique allows us to make quantitative comparisons between the performance of an information-theoretically optimal strategy with other candidate strategies on a food gathering task. We find that: 1. Information optimal search does not necessarily optimize utility (expected food gain). 2. The rank ordering of search strategies by information performance does not predict their ordering by expected food obtained. 3. The relative advantage of different strategies depends on the statistical structure of the environment, in particular the variability of motion of the source. We conclude that there is no simple relationship between information and utility. Behavioral optimality does not imply information efficiency, nor is there a simple tradeoff between gaining information about a food source versus obtaining the food itself., Comment: 52 pages, 14 figures