Control theory arose from a need to control synthetic systems. From regulating steam engines to tuning radios to devices capable of autonomous movement, it provided a formal mathematical basis for understanding the role of feedback in the stability (or change) of dynamical systems. It provides a framework for understanding any system with regulation via feedback, including biological ones such as regulatory gene networks, cellular metabolic systems, sensorimotor dynamics of moving animals, and even ecological or evolutionary dynamics of organisms and populations. Here, we focus on four case studies of the sensorimotor dynamics of animals, each of which involves the application of principles from control theory to probe stability and feedback in an organism's response to perturbations. We use examples from aquatic (two behaviors performed by electric fish), terrestrial (following of walls by cockroaches), and aerial environments (flight control by moths) to highlight how one can use control theory to understand the way feedback mechanisms interact with the physical dynamics of animals to determine their stability and response to sensory inputs and perturbations. Each case study is cast as a control problem with sensory input, neural processing, and motor dynamics, the output of which feeds back to the sensory inputs. Collectively, the interaction of these systems in a closed loop determines the behavior of the entire system.

The beauty of the JAR is that the behavioral response can be predicted based on a simple algorithm (Heiligenberg, 1991). For the fish to shift its EOD frequency in the 'correct' direction (e.g. the direction that increases |df |), the fish must be able to compute the sign of the df. The fish does this without an efference copy of its own EOD (Bullock et al., 1972) using amplitude and phase SUMMARY Recent studies have shown that central nervous system neurons in weakly electric fish respond to artificially constructed electrosensory envelopes, but the behavioral relevance of such stimuli is unclear. Here we investigate the possibility that social context creates envelopes that drive behavior. When Eigenmannia virescens are in groups of three or more, the interactions between their pseudo-sinusoidal electric fields can generate ʻsocial envelopesʼ. We developed a simple mathematical prediction for how fish might respond to such social envelopes. To test this prediction, we measured the responses of E. virescens to stimuli consisting of two sinusoids, each outside the range of the Jamming Avoidance Response (JAR), that when added to the fishʼs own electric field produced low-frequency (below 10Hz) social envelopes. Fish changed their electric organ discharge (EOD) frequency in response to these envelopes, which we have termed the Social Envelope Response (SER). In 99% of trials, the direction of the SER was consistent with the mathematical prediction. The SER was strongest in response to the lowest initial envelope frequency tested (2Hz) and depended on stimulus amplitude. The SER generally resulted in an increase of the envelope frequency during the course of a trial, suggesting that this behavior may be a mechanism for avoiding low-frequency social envelopes. Importantly, the direction of the SER was not predicted by the superposition of two JAR responses: the SER was insensitive to the amplitude ratio between the sinusoids used to generate the envelope, but was instead predicted by the sign of the difference of difference frequencies.

The study of animal behavior has been revolutionized by sophisticated methodologies that identify and track individuals in video recordings. Video recording of behavior, however, is challenging for many species and habitats including fishes that live in turbid water. Here we present a methodology for identifying and localizing weakly electric fishes on the centimeter scale with subsecond temporal resolution based solely on the electric signals generated by each individual. These signals are recorded with a grid of electrodes and analyzed using a two-part algorithm that identifies the signals from each individual fish and then estimates the position and orientation of each fish using Bayesian inference. Interestingly, because this system involves eavesdropping on electrocommunication signals, it permits monitoring of complex social and physical interactions in the wild. This approach has potential for large-scale non-invasive monitoring of aquatic habitats in the Amazon basin and other tropical freshwater systems.

13Hippocampal place cells are spatially tuned neurons that serve as elements of a 14 "cognitive map" in the mammalian brain 1 . To detect the animal's location, place cells 15 are thought to rely upon two interacting mechanisms: sensing the animal's position 16 relative to familiar landmarks 2,3 and measuring the distance and direction that the 17 animal has travelled from previously occupied locations 4-7 . The latter mechanism, 18 known as path integration, requires a finely tuned gain factor that relates the animal's 19 self-movement to the updating of position on the internal cognitive map, with external 20 landmarks necessary to correct positional error that eventually accumulates 8,9 . Path-21 integration-based models of hippocampal place cells and entorhinal grid cells treat 22the path integration gain as a constant 9-14 , but behavioral evidence in humans 23 suggests that the gain is modifiable 15 . Here we show physiological evidence from 24 hippocampal place cells that the path integration gain is indeed a highly plastic 25 variable that can be altered by persistent conflict between self-motion cues and 26 feedback from external landmarks. In a novel, augmented reality system, visual 27 landmarks were moved in proportion to the animal's movement on a circular track, 28 creating continuous conflict with path integration. Sustained exposure to this cue 29 conflict resulted in predictable and prolonged recalibration of the path integration 30 gain, as estimated from the place cells after the landmarks were extinguished. We 31 propose that this rapid plasticity keeps the positional update in register with the 32 60 Figure 1| Dome apparatus, experimental procedure, and sample data. a, Rendering of 61 dome apparatus. The dome shell is rendered semi-transparent for illustrative purposes. b, 62Photo of the apparatus. The dome is raised in the photo to allow visualization of the interior, 63but it is lowered as in (a) for the experiment. c, Illustration of experimental gain G. From the 64 same initial positions of the landmarks and rat, three different gain conditions are shown, in 65 both lab (top) and landmark (bottom) frames of reference. In each case, the rat runs 90° in 66 the lab frame. d, Profile of gain change and epochs during a typical session. An annular ring 67is always projected at the top of the dome (as shown in (a)) for illumination purposes, and is 68 not turned off even in Epoch 4. e, Representative firing rate maps for five different units from 69 five separate gain manipulation sessions, shown in the lab frame (top, middle rows) and 70 landmark frame (bottom row) during Epoch 3 (when the experimental gain was constant). 71 The plots in the top row are color scaled to their own individual maximum firing rates, 72 whereas the middle and bottom row plots are color scaled to the maximum firing rate of the 73 bottom plot of each pair. The difference in spatially averaged firing rates between landmark 74 and lab frames results from the distributed firing of the cells over the entire track in the l...

Summary Hippocampal place cells are spatially tuned neurons that serve as elements of a “cognitive map” in the mammalian brain 1 . To detect the animal’s location, place cells are thought to rely upon two interacting mechanisms: sensing the animal’s position relative to familiar landmarks 2 , 3 and measuring the distance and direction that the animal has traveled from previously occupied locations 4 – 7 . The latter mechanism, known as path integration , requires a finely tuned gain factor that relates the animal’s self-movement to the updating of position on the internal cognitive map, with external landmarks necessary to correct positional error that accumulates 8 , 9 . Path-integration-based models of hippocampal place cells and entorhinal grid cells treat the path integration gain as a constant 9 – 14 , but behavioral evidence in humans suggests that the gain is modifiable 15 . Here we show physiological evidence from hippocampal place cells that the path integration gain is indeed a highly plastic variable that can be altered by persistent conflict between self-motion cues and feedback from external landmarks. In a novel, augmented reality system, visual landmarks were moved in proportion to the animal’s movement on a circular track, creating continuous conflict with path integration. Sustained exposure to this cue conflict resulted in predictable and prolonged recalibration of the path integration gain, as estimated from the place cells after the landmarks were extinguished. We propose that this rapid plasticity keeps the positional update in register with the animal’s movement in the external world over behavioral timescales. These results also demonstrate that visual landmarks not only provide a signal to correct cumulative error in the path integration system 4 , 8 , 16 – 19 , but also rapidly fine-tune the integration computation itself.

SUMMARYThe Jamming Avoidance Response, or JAR, in the weakly electric fish has been analyzed at all levels of organization, from wholeorganism behavior down to specific ion channels. Nevertheless, a parsimonious description of the JAR behavior in terms of a dynamical system model has not been achieved at least in part due to the fact that 'avoidance' behaviors are both intrinsically unstable and nonlinear. We overcame the instability of the JAR in Eigenmannia virescens by closing a feedback loop around the behavioral response of the animal. Specifically, the instantaneous frequency of a jamming stimulus was tied to the fish's own electrogenic frequency by a feedback law. Without feedback, the fish's own frequency diverges from the stimulus frequency, but appropriate feedback stabilizes the behavior. After stabilizing the system, we measured the responses in the fish's instantaneous frequency to various stimuli. A delayed first-order linear system model fitted the behavior near the equilibrium. Coherence to white noise stimuli together with quantitative agreement across stimulus types supported this local linear model. Next, we examined the intrinsic nonlinearity of the behavior using clamped frequency difference experiments to extend the model beyond the neighborhood of the equilibrium. The resulting nonlinear model is composed of competing motor return and sensory escape terms. The model reproduces responses to step and ramp changes in the difference frequency (df) and predicts a 'snap-through' bifurcation as a function of dF that we confirmed experimentally.

Many biological phenomena such as locomotion, circadian cycles and breathing are rhythmic in nature and can be modelled as rhythmic dynamical systems. Dynamical systems modelling often involves neglecting certain characteristics of a physical system as a modelling convenience. For example, human locomotion is frequently treated as symmetric about the sagittal plane. In this work, we test this assumption by examining human walking dynamics around the steady state (limit-cycle). Here, we adapt statistical cross-validation in order to examine whether there are statistically significant asymmetries and, even if so, test the consequences of assuming bilateral symmetry anyway. Indeed, we identify significant asymmetries in the dynamics of human walking, but nevertheless show that ignoring these asymmetries results in a more consistent and predictive model. In general, neglecting evident characteristics of a system can be more than a modelling convenience-it can produce a better model.

Here, we review the role of control theory in modeling neural control systems through a top-down analysis approach. Specifically, we examine the role of the brain and central nervous system as the controller in the organism, connected to but isolated from the rest of the animal through insulated interfaces. Though biological and engineering control systems operate on similar principles, they differ in several critical features, which makes drawing inspiration from biology for engineering controllers challenging but worthwhile. We also outline a procedure that the control theorist can use to draw inspiration from the biological controller: starting from the intact, behaving animal; designing experiments to deconstruct and model hierarchies of feedback; modifying feedback topologies; perturbing inputs and plant dynamics; using the resultant outputs to perform system identification; and tuning and validating the resultant control-theoretic model using specially engineered robophysical models.