Margaret Trautner Department of Mathematics Sai Ravela † Department of Earth, Atmospheric, and Planetary Sciences Earth Signals and Systems Group, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: November 26, 2019) Neural dynamical systems are dynamical systems that are described at least in part by neural networks. The class of continuous-time neural dynamical systems must, however, be numerically integrated for simulation and learning. Modeled as recurrent networks embedding a continuous neural differential equation, they achieve fully neural temporal output. Using the polynomial class of dynamical systems, we demonstrate equivalence of neural and numerical integration. I. INTRODUCTION Neural dynamical systems are dynamical systems described at least in part by neural networks. Our interest in the subject emerges in the context of Systems Dynamics and Optimization [21] (SDO), which is central to many applications such as storm prediction [19], climate-risk based decision support [22], or autonomous observatories [25]. The SDO cycle conceptually involves a forward path dynamically parameterizing, reducing, calibrating, initializing and simulating numerical models, and quantifying their uncertainties. SDO further involves a return path for adaptive observation, inversion and estimation.

Integration in the head-direction system is a computation by which horizontal angular head velocity signals from the vestibular nuclei are integrated to yield a neural representation of head direction. In the thalamus, the postsubiculum and the mammillary nuclei, the head-direction representation has the form of a place code: neurons have a preferred head direction in which their firing is maximal [Blair and Sharp, 1995, Blair et al., 1998,?]. Integration is a difficult computation, given that head-velocities can vary over a large range. Previous models of the head-direction system relied on the assumption that the integration is achieved in a firing-rate-based attractor network with a ring structure. In order to correctly integrate head-velocity signals during high-speed head rotations, very fast synaptic dynamics had to be assumed. Here we address the question whether integration in the head-direction system is possible with slow synapses, for example excitatory NMDA and inhibitory GABA(B) type synapses. For neural networks with such slow synapses, rate-based dynamics are a good approximation of spiking neurons [Ermentrout, 1994]. We find that correct integration during high-speed head rotations imposes strong constraints on possible network architectures.

Integration in the head-direction system is a computation by which horizontal angular head velocity signals from the vestibular nuclei are integrated to yield a neural representation of head direction. In the thalamus, the postsubiculum and the mammillary nuclei, the head-direction representation has the form of a place code: neurons have a preferred head direction in which their firing is maximal [Blair and Sharp, 1995, Blair et al., 1998,?]. Integration is a difficult computation, given that head-velocities can vary over a large range. Previous models of the head-direction system relied on the assumption that the integration is achieved in a firing-rate-based attractor network with a ring structure. In order to correctly integrate head-velocity signals during high-speed head rotations, very fast synaptic dynamics had to be assumed. Here we address the question whether integration in the head-direction system is possible with slow synapses, for example excitatory NMDA and inhibitory GABA(B) type synapses. For neural networks with such slow synapses, rate-based dynamics are a good approximation of spiking neurons [Ermentrout, 1994]. We find that correct integration during high-speed head rotations imposes strong constraints on possible network architectures.

Integration in the head-direction system is a computation by which horizontal angularhead velocity signals from the vestibular nuclei are integrated toyield a neural representation of head direction. In the thalamus, the postsubiculum and the mammillary nuclei, the head-direction representation has the form of a place code: neurons have a preferred head direction in which their firing is maximal [Blair and Sharp, 1995, Blair et al., 1998,?]. Integration is a difficult computation, given that head-velocities can vary over a large range. Previous models of the head-direction system relied on the assumption that the integration is achieved in a firing-rate-based attractor network with a ring structure. In order to correctly integrate head-velocity signals during high-speed head rotations, very fast synaptic dynamics had to be assumed. Here we address the question whether integration in the head-direction system is possible with slow synapses, for example excitatory NMDA and inhibitory GABA(B) type synapses. For neural networks with such slow synapses, rate-based dynamics are a good approximation of spiking neurons[Ermentrout, 1994]. We find that correct integration during high-speed head rotations imposes strong constraints on possible network architectures.