Editing Walking (section)

== Mathematical models ==
Multiple mathematical models have been proposed to reproduce the kinematics observed in walking. These may be broadly broken down into four categories: rule-based models based on mechanical considerations and past literature, weakly coupled phase oscillators models, control-based models which guide simulations to maximize some property of locomotion, and phenomenological models which fit equations directly to the kinematics.

=== Rule-based models ===
The rule-based models integrate the past literature on motor control to generate a few simple rules which are presumed to be responsible for walking (e.g. “loading of the left leg triggers unloading of right leg”).<ref>{{Cite journal |last1=Schilling |first1=Malte |last2=Hoinville |first2=Thierry |last3=Schmitz |first3=Josef |last4=Cruse |first4=Holk |date=2013-07-04 |title=Walknet, a bio-inspired controller for hexapod walking |url=http://dx.doi.org/10.1007/s00422-013-0563-5 |journal=Biological Cybernetics |volume=107 |issue=4 |pages=397–419 |doi=10.1007/s00422-013-0563-5 |pmid=23824506 |pmc=3755227 |issn=0340-1200}}</ref><ref>{{Cite journal |last1=Geyer |first1=Hartmut |last2=Herr |first2=Hugh |date=June 2010 |title=A Muscle-Reflex Model That Encodes Principles of Legged Mechanics Produces Human Walking Dynamics and Muscle Activities |url=https://ieeexplore.ieee.org/document/5445011 |journal=IEEE Transactions on Neural Systems and Rehabilitation Engineering |volume=18 |issue=3 |pages=263–273 |doi=10.1109/TNSRE.2010.2047592 |pmid=20378480 |hdl=1721.1/70926 |s2cid=2041375 |issn=1558-0210|hdl-access=free }}</ref> Such models are generally most strictly based on the past literature and when they are based on a few rules can be easy to interpret. However, the influence of each rule can be hard to interpret when these models become more complex. Furthermore, the tuning of parameters is often done in an ad hoc way, revealing little intuition about why the system may be organized in this way. Finally, such models are typically based fully on sensory feedback, ignoring the effect of descending and rhythm generating neurons, which have been shown to be crucial in coordinating proper walking.

=== Coupled oscillator models ===
Dynamical system theory shows that any network with cyclical dynamics may be modeled as a set of [[Coupled oscillators|weakly coupled phase oscillators]], so another line of research has been exploring this view of walking.<ref>{{Cite journal |last1=Couzin-Fuchs |first1=Einat |last2=Kiemel |first2=Tim |last3=Gal |first3=Omer |last4=Ayali |first4=Amir |last5=Holmes |first5=Philip |date=2015-01-15 |title=Intersegmental coupling and recovery from perturbations in freely running cockroaches |url=http://dx.doi.org/10.1242/jeb.112805 |journal=Journal of Experimental Biology |volume=218 |issue=2 |pages=285–297 |doi=10.1242/jeb.112805 |pmid=25609786 |pmc=4302167 |bibcode=2015JExpB.218..285C |issn=1477-9145}}</ref> Each oscillator may model a muscle, joint angle, or even a whole leg, and is coupled to some set of other oscillators. Often, these oscillators are thought to represent the [[central pattern generator]]s underlying walking. These models have rich theory behind them, allow for some extensions based on sensory feedback, and can be fit to kinematics. However, they need to be heavily constrained to fit to data and by themselves make no claims on which gaits allow the animal to move faster, more robustly, or more efficiently.

=== Control based models ===
Control-based models start with a simulation based on some description of the animal's anatomy and optimize control parameters to generate some behavior. These may be based on a musculoskeletal model,<ref>{{Cite journal |last1=Geijtenbeek |first1=Thomas |last2=van de Panne |first2=Michiel |last3=van der Stappen |first3=A. Frank |date=November 2013 |title=Flexible muscle-based locomotion for bipedal creatures |url=http://dx.doi.org/10.1145/2508363.2508399 |journal=ACM Transactions on Graphics |volume=32 |issue=6 |pages=1–11 |doi=10.1145/2508363.2508399 |s2cid=9183862 |issn=0730-0301}}</ref> skeletal model,<ref>{{cite arXiv | eprint=1707.02286 | last1=Heess | first1=Nicolas | last2=TB | first2=Dhruva | last3=Sriram | first3=Srinivasan | last4=Lemmon | first4=Jay | last5=Merel | first5=Josh | last6=Wayne | first6=Greg | last7=Tassa | first7=Yuval | last8=Erez | first8=Tom | last9=Wang | first9=Ziyu | last10=Ali Eslami | first10=S. M. | last11=Riedmiller | first11=Martin | last12=Silver | first12=David | title=Emergence of Locomotion Behaviours in Rich Environments | year=2017 | class=cs.AI }}</ref><ref>{{Cite journal |last1=Peng |first1=Xue Bin |last2=Abbeel |first2=Pieter |last3=Levine |first3=Sergey |last4=van de Panne |first4=Michiel |date=2018-08-31 |title=DeepMimic |url=http://dx.doi.org/10.1145/3197517.3201311 |journal=ACM Transactions on Graphics |volume=37 |issue=4 |pages=1–14 |doi=10.1145/3197517.3201311 |arxiv=1804.02717 |s2cid=215808400 |issn=0730-0301}}</ref> or even simply a ball and stick model.<ref>{{Cite journal |last1=Szczecinski |first1=Nicholas S. |last2=Bockemühl |first2=Till |last3=Chockley |first3=Alexander S. |last4=Büschges |first4=Ansgar |date=2018-11-16 |title=Static stability predicts the continuum of interleg coordination patterns in Drosophila |journal=Journal of Experimental Biology |volume=221 |issue=22 |pages=jeb189142 |doi=10.1242/jeb.189142 |pmid=30274987 |s2cid=52903595 |issn=0022-0949|doi-access=free }}</ref> As these models generate locomotion by optimizing some metric, they can be used to explore the space of optimal locomotion behaviors under some assumptions. However, they typically do not generate plausible hypotheses on the neural coding underlying the behaviors and are typically sensitive to modeling assumptions.

=== Statistical models ===
Phenomenological models model the kinematics of walking directly by fitting a [[dynamical system]], without postulating an underlying mechanism for how the kinematics are generated neurally. Such models can produce the most realistic kinematic trajectories and thus have been explored for simulating walking for [[Computer animation|computer-based animation]].<ref>{{Cite journal |last1=Holden |first1=Daniel |last2=Komura |first2=Taku |last3=Saito |first3=Jun |date=2017-07-20 |title=Phase-functioned neural networks for character control |url=https://doi.org/10.1145/3072959.3073663 |journal=ACM Transactions on Graphics |volume=36 |issue=4 |pages=42:1–42:13 |doi=10.1145/3072959.3073663 |hdl=20.500.11820/c09514d6-427f-4e00-adcc-1466f0125135 |s2cid=7261259 |issn=0730-0301|hdl-access=free }}</ref><ref>{{Cite journal |last1=Zhang |first1=He |last2=Starke |first2=Sebastian |last3=Komura |first3=Taku |last4=Saito |first4=Jun |date=2018-07-30 |title=Mode-adaptive neural networks for quadruped motion control |url=https://doi.org/10.1145/3197517.3201366 |journal=ACM Transactions on Graphics |volume=37 |issue=4 |pages=145:1–145:11 |doi=10.1145/3197517.3201366 |s2cid=51692385 |issn=0730-0301}}</ref> However, the lack of underlying mechanism makes it hard to apply these models to study the biomechanical or neural properties of walking.