Learning and recall in a dynamic theory of coordination patterns

A dynamic theory of learning and recall of coordination patterns is developed in the context of relative timing skills. Characterizing the coordination patterns in such skills by the collective variable, relative phase, we choose a model system in which the intrinsic pattern dynamics as well as the influence of environmental and memorized information are well understood from previous experimental and theoretical work. To describe learning we endow memorized information with dynamics which is determined by a phenomenological strategy. Similarly, additional degrees of freedom must be introduced to understand recall. As such recall variables we choose the relative strengths with which each memorized pattern acts on the pattern dynamics and model their dynamics phenomenologically. The resulting dynamical system that resembles models used in pattern recognition theory is shown to adequately describe the learning and recall processes. Moreover, due to the operational character of the theory, several predictions emerge that are open to experimental test. In particular, we show under which conditions phase transitions occur in the dynamics of the coordination patterns during learning and during recall. Considering different time scales and their relations we demonstrate how these phase transitions can be identified and observed. Other predictions include the influence of the intrinsic pattern dynamics on the recall process and the existence of history and hysteresis effects in recall. We discuss different forms of forgetting and differentiation of memorized information. The results show how a new theoretical view of learning and recall as change of behavioral dynamics can lead to a different understanding of these processes by providing testable predictions.     

Morphology and the gradient of a symmetric potential predict gait transitions of dogs

Gaits and gait transitions play a central role in the movement of animals. Symmetry is thought to govern the structure of the nervous system, and constrain the limb motions of quadrupeds. We quantify the symmetry of dog gaits with respect to combinations of bilateral, fore–aft, and spatio-temporal symmetry groups. We tested the ability of symmetries to model motion capture data of dogs walking, trotting and transitioning between those gaits. Fully symmetric models performed comparably to asymmetric with only a \(22\%\) increase in the residual sum of squares and only one-quarter of the parameters. This required adding a spatio-temporal shift representing a lag between fore and hind limbs. Without this shift, the symmetric model residual sum of squares was \(1700\%\) larger. This shift is related to (linear regression, \(n=5\), \(p=0.0328\)) dog morphology. That this symmetry is respected throughout the gaits and transitions indicates that it generalizes outside a single gait. We propose that relative phasing of limb motions can be described by an interaction potential with a symmetric structure. This approach can be extended to the study of interaction of neurodynamic and kinematic variables, providing a system-level model that couples neuronal central pattern generator networks and mechanical models.     

Models of central pattern generators for quadruped locomotion II. Secondary gaits

We continue the analysis of the network of symmetrically coupled cells modeling central pattern generators (CPG) for quadruped locomotion proposed by Golubitsky, Stewart, Buono and Collins by studying secondary gaits. Secondary gaits are modeled by output signals from the CPG where each cell emits one of two different output signals along with exact phase shifts. Examples of secondary gaits are transverse gallop, rotary gallop, and canter. We classify secondary gaits that bifurcate when the Poincaré map of a primary gait has a real eigenvalue crossing the unit circle. In particular, we show that periodic solutions modeling transverse gallop and rotary gallop bifurcate from primary gaits. Moreover, we find gaits from period-doubling bifurcations and analyze plausible footfall patterns. Numerical simulations are performed using the Morris-Lecar equations as cell dynamics.     

Hard-wired central pattern generators for quadrupedal locomotion

Animal locomotion is generated and controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of producing rhythmic output. In the present work, it is demonstrated that a hard-wired CPG model, made up of four coupled nonlinear oscillators, can produce multiple phase-locked oscillation patterns that correspond to three common quadrupedal gaits — the walk, trot, and bound. Transitions between the different gaits are generated by varying the network's driving signal and/or by altering internal oscillator parameters. The above in numero results are obtained without changing the relative strengths or the polarities of the system's synaptic interconnections, i.e., the network maintains an invariant coupling architecture. It is also shown that the ability of the hard-wired CPG network to produce and switch between multiple gait patterns is a model-independent phenomenon, i.e., it does not depend upon the detailed dynamics of the component oscillators and/or the nature of the inter-oscillator coupling. Three different neuronal oscillator models — the Stein neuronal model, the Van der Pol oscillator, and the FitzHugh-Nagumo model -and two different coupling schemes are incorporated into the network without impeding its ability to produce the three quadrupedal gaits and the aforementioned gait transitions.     

Relative spike timing in stochastic oscillator networks of the Hermissenda eye

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye.     

Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects

 Motivated by experimental studies of insects, we propose a model for legged locomotion in the horizontal plane. A three-degree-of freedom, energetically conservative, rigid-body model with a pair of compliant virtual legs in intermittent contact with the ground allows us to study how dynamics depends on parameters such as mass, moment of inertia, leg stiffness, and length. We find periodic gaits, and show that mechanics alone can confer asymptotic stability of relative heading and body angular velocity. We discuss the relevance of our idealized models to experiments and simulations on insect running, showing that their gait and force characteristics match observations reasonably well. We perform parameter studies and suggest that our model is relevant to the understanding of locomotion dynamics across species. Received: 17 April 2001 / Accepted in revised form: 20 November 2001     

Financial Symmetry and Moods in the Market

This paper studies how certain speculative transitions in financial markets can be ascribed to a symmetry break that happens in the collective decision making. Investors are assumed to be bounded rational, using a limited set of information including past price history and expectation on future dividends. Investment strategies are dynamically changed based on realized returns within a game theoretical scheme with Nash equilibria. In such a setting, markets behave as complex systems whose payoff reflect an intrinsic financial symmetry that guarantees equilibrium in price dynamics (fundamentalist state) until the symmetry is broken leading to bubble or anti-bubble scenarios (speculative state). We model such two-phase transition in a micro-to-macro scheme through a Ginzburg-Landau-based power expansion leading to a market temperature parameter which modulates the state transitions in the market. Via simulations we prove that transitions in the market price dynamics can be phenomenologically explained by the number of traders, the number of strategies and amount of information used by agents, all included in our market temperature parameter.     

Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation

In order to assess the relative contributions to pattern-generation of the intrinsic properties of individual neurons and of their connectivity, we examined a ring circuit composed of four complex physiologically based oscillators. This circuit produced patterns that correspond to several quadrupedal gaits, including the walk, the bound, and the gallop. An analysis using the phase response curve (PRC) of an uncoupled oscillator accurately predicted all modes exhibited by this circuit and their phasic relationships – with the caveat that in certain parameter ranges, bistability in the individual oscillators added nongait patterns that were not amenable to PRC analysis, but further enriched the pattern-generating repertoire of the circuit. The key insights in the PRC analysis were that in a gait pattern, since all oscillators are entrained at the same frequency, the phase advance or delay caused by the action of each oscillator on its postsynaptic oscillator must be the same, and the sum of the normalized phase differences around the ring must equal to an integer. As suggested by several previous studies, our analysis showed that the capacity to exhibit a large number of patterns is inherent in the ring circuit configuration. In addition, our analysis revealed that the shape of the PRC for the individual oscillators determines which of the theoretically possible modes can be generated using these oscillators as circuit elements. PRCs that have a complex shape enable a circuit to produce a wider variety of patterns, and since complex neurons tend to have complex PRCs, enriching the repertoire of patterns exhibited by a circuit may be the function of some intrinsic neuronal complexity. Our analysis showed that gait transitions, or more generally, pattern transitions, in a ring circuit do not require rewiring the circuit or any changes in the strength of the connections. Instead, transitions can be achieved by using a control parameter, such as stimulus intensity, to sculpt the PRC so that it has the appropriate shape for the desired pattern(s). A transition can then be achieved simply by changing the value of the control parameter so that the first pattern either ceases to exist or loses stability, while a second pattern either comes into existence or gains stability. Our analysis illustrates the predictive value of PRCs in circuit analysis and can be extended to provide a design method for pattern-generating circuits. Received: 20 November 1996 / Accepted: 29 July 1997     

Human odometer is gait-symmetry specific

In 1709, Berkeley hypothesized of the human that distance is measurable by ‘the motion of his body, which is perceivable by touch’. To be sufficiently general and reliable, Berkeley''s hypothesis must imply that distance measured by legged locomotion approximates actual distance, with the measure invariant to gait, speed and number of steps. We studied blindfolded human participants in a task in which they travelled by legged locomotion from a fixed starting point A to a variable terminus B, and then reproduced, by legged locomotion from B, the A–B distance. The outbound (‘measure’) and return (‘report’) gait could be the same or different, with similar or dissimilar step sizes and step frequencies. In five experiments we manipulated bipedal gait according to the primary versus secondary distinction revealed in symmetry group analyses of locomotion patterns. Berkeley''s hypothesis held only when the measure and report gaits were of the same symmetry class, indicating that idiothetic distance measurement is gait-symmetry specific. Results suggest that human odometry (and perhaps animal odometry more generally) entails variables that encompass the limbs in coordination, such as global phase, and not variables at the level of the single limb, such as step length and step number, as traditionally assumed.     

Leg recirculation in horizontal plane locomotion

A protocol prescribing leg motion during the swing phase is developed for the planar lateral leg spring model of locomotion. Inspired by experimental observations regarding insect leg function when running over rough terrain, the protocol prescribes the angular velocity of the swing-leg relative to the body in a feedforward manner, yielding natural variations in the leg touch-down angle in response to perturbations away from a periodic orbit. Analysis of the reduced order model reveals that periodic gait stability and robustness to external perturbations depends strongly upon the angular velocity of the leg at touch-down. While the leg angular velocity at touch-down provides control over gait stability and can be chosen to stabilize unstable gaits, the resulting basin of stability is much smaller than that observed for the original lateral leg spring model with a fixed leg touch-down angle. Comparisons to experimental leg angular velocity data for running cockroaches reveal that while the proposed protocol is qualitatively correct, smaller leg angular accelerations occur during the second half of the swing phase. Modifications made to the recirculation protocol to better match experimental observations yield large improvements in the basin of stability.     

Angular momentum and arboreal stability in common marmosets (Callithrix jacchus)

Despite the importance that concepts of arboreal stability have in theories of primate locomotor evolution, we currently lack measures of balance performance during primate locomotion. We provide the first quantitative data on locomotor stability in an arboreal primate, the common marmoset (Callithrix jacchus), predicting that primates should maximize arboreal stability by minimizing side-to-side angular momentum about the support (i.e., L_sup). If net L_sup becomes excessive, the animal will be unable to arrest its angular movement and will fall. Using a novel, highly integrative experimental procedure we directly measured whole-body L_sup in two adult marmosets moving along narrow (2.5 cm diameter) and broad (5 cm diameter) poles. Marmosets showed a strong preference for asymmetrical gaits (e.g., gallops and bounds) over symmetrical gaits (e.g., walks and runs), with asymmetrical gaits representing >90% of all strides. Movement on the narrow support was associated with an increase in more “grounded” gaits (i.e., lacking an aerial phase) and a more even distribution of torque production between the fore- and hind limbs. These adjustments in gait dynamics significantly reduced net L_sup on the narrow support relative to the broad support. Despite their lack of a well-developed grasping apparatus, marmosets proved adept at producing muscular “grasping” torques about the support, particularly with the hind limbs. We contend that asymmetrical gaits permit small-bodied arboreal mammals, including primates, to expand “effective grasp” by gripping the substrate between left and right limbs of a girdle. This model of arboreal stability may hold important implications for understanding primate locomotor evolution. Am J Phys Anthropol 156:565–576, 2015. © 2014 Wiley Periodicals, Inc.     

Substrate determines asymmetrical gait dynamics in marmosets (Callithrix jacchus) and squirrel monkeys (Saimiri boliviensis)

Studies of skeletal pathology indicate that injury from falling accounts for most long bone trauma in free‐ranging primates, suggesting that primates should be under strong selection to manifest morphological and behavioral mechanisms that increase stability on arboreal substrates. Although previous studies have identified several kinematic and kinetic features of primate symmetrical gaits that serve to increase arboreal stability, very little work has focused on the dynamics of primate asymmetrical gaits. Nevertheless, asymmetrical gaits typify the rapid locomotion of most primates, particularly in smaller bodied taxa. This study investigated asymmetrical gait dynamics in growing marmosets and squirrel monkeys moving on terrestrial and simulated arboreal supports (i.e., an elevated pole). Results showed that monkeys used several kinematic and kinetic adjustments to increase stability on the pole, including reducing peak vertical forces, limiting center of mass movements, increasing substrate contact durations, and using shorter and more frequent strides (thus limiting disruptive whole‐body aerial phases). Marmosets generally showed greater adjustment to pole locomotion than did squirrel monkeys, perhaps as a result of their reduced grasping abilities and retreat from the fine‐branch niche. Ontogenetic increases in body size had relatively little independent influence on asymmetrical gait dynamics during pole locomotion, despite biomechanical theory suggesting that arboreal instability is exacerbated as body size increases relative to substrate diameter. Overall, this study shows that 1) symmetrical gaits are not the only stable way to travel arboreally and 2) small‐bodied primates utilize specific kinematic and kinetic adjustments to increase stability when using asymmetrical gaits on arboreal substrates. Am J Phys Anthropol, 2009. © 2008 Wiley‐Liss, Inc.     

Hexapodal gaits and coupled nonlinear oscillator models

The general, model-independent features of different networks of six symmetrically coupled nonlinear oscillators are investigated. These networks are considered as possible models for locomotor central pattern generators (CPGs) in insects. Numerical experiments with a specific oscillator network model are briefly described. It is shown that some generic phase-locked oscillation-patterns for various systems of six symmetrically coupled nonlinear oscillators correspond to the common forward-walking gaits adopted by insects. It is also demonstrated that transitions observed in insect gaits can be modelled as standard symmetry-breaking bifurcations occurring in such systems. The present analysis, which leads to a natural classification of hexapodal gaits by symmetry and to natural sequences of gait bifurcations, relates observed gaits to the overall organizational structure of the underlying CPG. The implications of the present results for the development of simplified control systems for hexapodal walking robots are discussed.     

Patterns of mechanical energy change in tetrapod gait: pendula, springs and work

Kinematic and center of mass (CoM) mechanical variables used to define terrestrial gaits are compared for various tetrapod species. Kinematic variables (limb phase, duty factor) provide important timing information regarding the neural control and limb coordination of various gaits. Whereas, mechanical variables (potential and kinetic energy relative phase, %Recovery, %Congruity) provide insight into the underlying mechanisms that minimize muscle work and the metabolic cost of locomotion, and also influence neural control strategies. Two basic mechanisms identified by Cavagna et al. (1977. Am J Physiol 233:R243-R261) are used broadly by various bipedal and quadrupedal species. During walking, animals exchange CoM potential energy (PE) with kinetic energy (KE) via an inverted pendulum mechanism to reduce muscle work. During the stance period of running (including trotting, hopping and galloping) gaits, animals convert PE and KE into elastic strain energy in spring elements of the limbs and trunk and regain this energy later during limb support. The bouncing motion of the body on the support limb(s) is well represented by a simple mass-spring system. Limb spring compliance allows the storage and return of elastic energy to reduce muscle work. These two distinct patterns of CoM mechanical energy exchange are fairly well correlated with kinematic distinctions of limb movement patterns associated with gait change. However, in some cases such correlations can be misleading. When running (or trotting) at low speeds many animals lack an aerial period and have limb duty factors that exceed 0.5. Rather than interpreting this as a change of gait, the underlying mechanics of the body's CoM motion indicate no fundamental change in limb movement pattern or CoM dynamics has occurred. Nevertheless, the idealized, distinctive patterns of CoM energy fluctuation predicted by an inverted pendulum for walking and a bouncing mass spring for running are often not clear cut, especially for less cursorial species. When the kinematic and mechanical patterns of a broader diversity of quadrupeds and bipeds are compared, more complex patterns emerge, indicating that some animals may combine walking and running mechanics at intermediate speeds or at very large size. These models also ignore energy costs that are likely associated with the opposing action of limbs that have overlapping support times during walking. A recent model of terrestrial gait (Ruina et al., 2005. J Theor Biol, in press) that treats limb contact with the ground in terms of collisional energy loss indicates that considerable CoM energy can be conserved simply by matching the path of CoM motion perpendicular to limb ground force. This model, coupled with the earlier ones of pendular exchange during walking and mass-spring elastic energy savings during running, provides compelling argument for the view that the legged locomotion of quadrupeds and other terrestrial animals has generally evolved to minimize muscle work during steady level movement.     

Gaits of Japanese macaques (Macaca fuscata) on a horizontal ladder and arboreal stability

Most primates use diagonal sequence (DS), diagonal couplets (DC) gaits when they walk or run quadrupedally, and it has been suggested that DSDC gaits contribute to stability in their natural arboreal habitats compared to other symmetrical gaits. However, this postulate is based solely on studies of primate gaits using continuous terrestrial and arboreal substrates. A particular species may select suitable gaits according to the substrate properties. Here, we analyzed the gaits of Japanese macaques moving on a horizontal ladder with rung intervals ranging from 0.40 to 0.80 m to elucidate the relative advantages of each observed form of gait. The rung arrangement forced our macaques to choose either diagonal coupling or DS gaits. One macaque consistently used diagonal coupling (i.e., DSDC and LSDC gaits) across narrow and intermediate rung intervals, whereas the other macaque used DS gaits (i.e., DSDC and DSLC gaits). At wider rung intervals, both macaques shifted to a two‐one sequence (TOS), which is characterized by two nearly simultaneous touchdowns of both forelimbs and one touchdown of each hind limb in a stride. The transition to the TOS sequence increased the duration of support on multiple limbs, but always included periods of a whole‐body aerial phase. These results suggest that Japanese macaques prefer DSDC gaits, because the diagonal coupling and DS contribute separately to stability on complex supports compared to the lateral coupling and lateral sequence. We also postulate that stability triggers the transition from symmetrical gaits to the TOS sequence. Am J Phys Anthropol, 2009. © 2008 Wiley‐Liss, Inc.     

The metabolic and mechanical costs of step time asymmetry in walking

Animals use both pendular and elastic mechanisms to minimize energy expenditure during terrestrial locomotion. Elastic gaits can be either bilaterally symmetric (e.g. run and trot) or asymmetric (e.g. skip, canter and gallop), yet only symmetric pendular gaits (e.g. walk) are observed in nature. Does minimizing metabolic and mechanical power constrain pendular gaits to temporal symmetry? We measured rates of metabolic energy expenditure and calculated mechanical power production while healthy humans walked symmetrically and asymmetrically at a range of step and stride times. We found that walking with a 42 per cent step time asymmetry required 80 per cent (2.5 W kg⁻¹) more metabolic power than preferred symmetric gait. Positive mechanical power production increased by 64 per cent (approx. 0.24 W kg⁻¹), paralleling the increases we observed in metabolic power. We found that when walking asymmetrically, subjects absorbed more power during double support than during symmetric walking and compensated by increasing power production during single support. Overall, we identify inherent metabolic and mechanical costs to gait asymmetry and find that symmetry is optimal in healthy human walking.     

From physics to biology by extending criticality and symmetry breakings

Symmetries play a major role in physics, in particular since the work by E. Noether and H. Weyl in the first half of last century. Herein, we briefly review their role by recalling how symmetry changes allow to conceptually move from classical to relativistic and quantum physics. We then introduce our ongoing theoretical analysis in biology and show that symmetries play a radically different role in this discipline, when compared to those in current physics. By this comparison, we stress that symmetries must be understood in relation to conservation and stability properties, as represented in the theories. We posit that the dynamics of biological organisms, in their various levels of organization, are not “just” processes, but permanent (extended, in our terminology) critical transitions and, thus, symmetry changes. Within the limits of a relative structural stability (or interval of viability), variability is at the core of these transitions.     

Kinematic control of walking

The planar law of inter-segmental co-ordination we described may emerge from the coupling of neural oscillators between each other and with limb mechanical oscillators. Muscle contraction intervenes at variable times to re-excite the intrinsic oscillations of the system when energy is lost. The hypothesis that a law of coordinative control results from a minimal active tuning of the passive inertial and viscoelastic coupling among limb segments is congruent with the idea that movement has evolved according to minimum energy criteria (1, 8). It is known that multi-segment motion of mammals locomotion is controlled by a network of coupled oscillators (CPGs, see 18, 33, 37). Flexible combination of unit oscillators gives rise to different forms of locomotion. Inter-oscillator coupling can be modified by changing the synaptic strength (or polarity) of the relative spinal connections. As a result, unit oscillators can be coupled in phase, out of phase, or with a variable phase, giving rise to different behaviors, such as speed increments or reversal of gait direction (from forward to backward). Supra-spinal centers may drive or modulate functional sets of coordinating interneurons to generate different walking modes (or gaits). Although it is often assumed that CPGs control patterns of muscle activity, an equally plausible hypothesis is that they control patterns of limb segment motion instead (22). According to this kinematic view, each unit oscillator would directly control a limb segment, alternately generating forward and backward oscillations of the segment. Inter-segmental coordination would be achieved by coupling unit oscillators with a variable phase. Inter-segmental kinematic phase plays the role of global control variable previously postulated for the network of central oscillators. In fact, inter-segmental phase shifts systematically with increasing speed both in man (4) and cat (38). Because this phase-shift is correlated with the net mechanical power output over a gait cycle (3, 4), phase control could be used for limiting the overall energy expenditure with increasing speed (22). Adaptation to different walking conditions, such as changes in body posture, body weight unloading and backward walk, also involves inter-segmental phase tuning, as does the maturation of limb kinematics in toddlers.     

A bipedal compliant walking model generates periodic gait cycles with realistic swing dynamics

A simple spring mechanics model can capture the dynamics of the center of mass (CoM) during human walking, which is coordinated by multiple joints. This simple spring model, however, only describes the CoM during the stance phase, and the mechanics involved in the bipedality of the human gait are limited. In this study, a bipedal spring walking model was proposed to demonstrate the dynamics of bipedal walking, including swing dynamics followed by the step-to-step transition. The model consists of two springs with different stiffnesses and rest lengths representing the stance leg and swing leg. One end of each spring has a foot mass, and the other end is attached to the body mass. To induce a forward swing that matches the gait phase, a torsional hip joint spring was introduced at each leg. To reflect the active knee flexion for foot clearance, the rest length of the swing leg was set shorter than that of the stance leg, generating a discrete elastic restoring force. The number of model parameters was reduced by introducing dependencies among stiffness parameters. The proposed model generates periodic gaits with dynamics-driven step-to-step transitions and realistic swing dynamics. While preserving the mimicry of the CoM and ground reaction force (GRF) data at various gait speeds, the proposed model emulated the kinematics of the swing leg. This result implies that the dynamics of human walking generated by the actuations of multiple body segments is describable by a simple spring mechanics.