Philip Holmes | Princeton University (original) (raw)

Papers by Philip Holmes

Research paper thumbnail of Dynamical Systems and Bifurcations of Vector Fields

Research paper thumbnail of Mechanical models for insect locomotion: active muscles and energy losses

Biological cybernetics, 2003

We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion ... more We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion in the horizontal plane developed in Schmitt and Holmes (2000a,b, 2001), where gaits characteristic of steady cockroach running and turning were evoked. In this paper, we include dissipation and energy inputs via active "muscles" in three forms: via prescribed torques at the "hip" pivot, via an active spring element of variable length, and via a pair of Hill-type muscle models representing an extensor/flexor system. Due to mechanical feedback of passive elastic forces, the stable gaits of the conservative models are preserved, and now energy input and absorption balances to additionally stabilize a preferred speed, with only modest neural sensing and feedback being required. However, these bipedal models still cannot simultaneously match observed moment-yaw magnitudes and fore-aft dynamics.

Research paper thumbnail of Nonlinear Propagation of Light in One-Dimensional Periodic Structures

Research paper thumbnail of Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model

Research paper thumbnail of Mechanical models for insect locomotion: stability and parameter studies

Research paper thumbnail of The physics of optimal decision making: A formal analysis of models of performance in two alternative forced choice tasks

In this article, the authors consider optimal decision making in two-alternative forced-choice (T... more In this article, the authors consider optimal decision making in two-alternative forced-choice (TAFC) tasks. They begin by analyzing 6 models of TAFC decision making and show that all but one can be reduced to the drift diffusion model, implementing the statistically optimal algorithm (most accurate for a given speed or fastest for a given accuracy). They prove further that there is always an optimal trade-off between speed and accuracy that maximizes various reward functions, including reward rate (percentage of correct responses per unit time), as well as several other objective functions, including ones weighted for accuracy. They use these findings to address empirical data and make novel predictions about performance under optimality.

Research paper thumbnail of Globally Coupled Oscillator Networks

We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimens... more We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimensional tori. We focus on the efiects of symmetry and of the forms of the coupling functions, derived from un- derlying Hodgkin-Huxley type neuron models, on the existence, stability, and degeneracy of phase-locked solutions in which subgroups of oscillators share common phases. We also estimate domains of attraction for

Research paper thumbnail of Simple Neural Networks that Optimize Decisions

International Journal of Bifurcation and Chaos, 2005

We review simple connectionist and flring rate models for mutually inhibiting pools of neurons th... more We review simple connectionist and flring rate models for mutually inhibiting pools of neurons that discriminate between pairs of stimuli. Both are two-dimensional nonlinear stochastic ordinary difierential equations, and although they difier in how inputs and stim- uli enter, we show that they are equivalent under state variable and parameter coordinate changes. A key parameter is gain: the maximum slope

Research paper thumbnail of Optimality and some of its discontents: successes and shortcomings of existing models for binary decisions

Topics in cognitive science, 2014

We review how leaky competing accumulators (LCAs) can be used to model decision making in two-alt... more We review how leaky competing accumulators (LCAs) can be used to model decision making in two-alternative, forced-choice tasks, and we show how they reduce to drift diffusion (DD) processes in special cases. As continuum limits of the sequential probability ratio test, DD processes are optimal in producing decisions of specified accuracy in the shortest possible time. Furthermore, the DD model can be used to derive a speed-accuracy trade-off that optimizes reward rate for a restricted class of two alternative forced-choice decision tasks. We review findings that compare human performance with this benchmark, and we reveal both approximations to and deviations from optimality. We then discuss three potential sources of deviations from optimality at the psychological level--avoidance of errors, poor time estimation, and minimization of the cost of control--and review recent theoretical and empirical findings that address these possibilities. We also discuss the role of cognitive contr...

Research paper thumbnail of The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

Siam Review, 2006

Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy... more Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal

Research paper thumbnail of Lateralized Readiness Potentials Reveal Properties of a Neural Mechanism for Implementing a Decision Threshold

PLoS ONE, 2014

Many perceptual decision making models posit that participants accumulate noisy evidence over tim... more Many perceptual decision making models posit that participants accumulate noisy evidence over time to improve the accuracy of their decisions, and that in free response tasks, participants respond when the accumulated evidence reaches a decision threshold. Research on the neural correlates of these models' components focuses primarily on evidence accumulation. Far less attention has been paid to the neural correlates of decision thresholds, reflecting the final commitment to a decision. Inspired by a model of bistable neural activity that implements a decision threshold, we reinterpret human lateralized readiness potentials (LRPs) as reflecting the crossing of a decision threshold. Interestingly, this threshold crossing preserves signatures of a drift-diffusion process of evidence accumulation that feeds in to the threshold mechanism. We show that, as our model predicts, LRP amplitudes and growth rates recorded while participants performed a motion discrimination task correlate with individual differences in behaviorally-estimated prior beliefs, decision thresholds and evidence accumulation rates. As such LRPs provide a useful measure to test dynamical models of both evidence accumulation and decision commitment processes non-invasively.

Research paper thumbnail of A comparative study of drift diffusion and linear ballistic accumulator models in a reward maximization perceptual choice task

Frontiers in neuroscience, 2014

We present new findings that distinguish drift diffusion models (DDMs) from the linear ballistic ... more We present new findings that distinguish drift diffusion models (DDMs) from the linear ballistic accumulator (LBA) model as descriptions of human behavior in a two-alternative forced-choice reward maximization (Rmax) task. Previous comparisons have not considered Rmax tasks, and differences identified between the models' predictions have centered on practice effects. Unlike the parameter-free optimal performance curves of the pure DDM, the extended DDM and LBA predict families of curves depending on their additional parameters, and those of the LBA show significant differences from the DDMs, especially for poorly discriminable stimuli that incur high error rates. Moreover, fits to behavior reveal that the LBA and DDM provide different interpretations of behavior as stimulus discriminability increases. Trends for threshold setting (caution) in the DDMs are consistent between fits, while in the corresponding LBA fits, thresholds interact with distributions of starting points in a ...

Research paper thumbnail of Simple networks that optimize decisions

International Journal of Bifurcation and Chaos - IJBC, 2004

Research paper thumbnail of Heteroclinic cycles and modulated traveling waves in systems with O(2) symmetry

Research paper thumbnail of The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

Research paper thumbnail of Kuramoto–Sivashinsky Dynamics on the Center–Unstable Manifold

SIAM Journal on Applied Mathematics, 1989

Research paper thumbnail of A Simply Stabilized Running Model

SIAM Journal on Applied Dynamical Systems, 2003

Research paper thumbnail of Can Monkeys Choose Optimally When Faced with Noisy Stimuli and Unequal Rewards?

PLoS Computational Biology, 2009

Research paper thumbnail of Nonlinear Muscles, Passive Viscoelasticity and Body Taper Conspire To Create Neuromechanical Phase Lags in Anguilliform Swimmers

PLoS Computational Biology, 2008

Locomotion provides superb examples of cooperation among neuromuscular systems, environmental rea... more Locomotion provides superb examples of cooperation among neuromuscular systems, environmental reaction forces, and sensory feedback. As part of a program to understand the neuromechanics of locomotion, here we construct a model of anguilliform (eel-like) swimming in slender fishes. Building on a continuum mechanical representation of the body as an viscoelastic rod, actuated by a traveling wave of preferred curvature and subject to hydrodynamic reaction forces, we incorporate a new version of a calcium release and muscle force model, fitted to data from the lamprey Ichthyomyzon unicuspis, that interactively generates the curvature wave. We use the model to investigate the source of the difference in speeds observed between electromyographic waves of muscle activation and mechanical waves of body curvature, concluding that it is due to a combination of passive viscoelastic and geometric properties of the body and active muscle properties. Moreover, we find that nonlinear force dependence on muscle length and shortening velocity may reduce the work done by the swimming muscles in steady swimming.

Research paper thumbnail of Strong NLS soliton–defect interactions

Physica D: Nonlinear Phenomena, 2004

Research paper thumbnail of Dynamical Systems and Bifurcations of Vector Fields

Research paper thumbnail of Mechanical models for insect locomotion: active muscles and energy losses

Biological cybernetics, 2003

We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion ... more We extend the analysis of simple, energy-conserving models for the dynamics of insect locomotion in the horizontal plane developed in Schmitt and Holmes (2000a,b, 2001), where gaits characteristic of steady cockroach running and turning were evoked. In this paper, we include dissipation and energy inputs via active "muscles" in three forms: via prescribed torques at the "hip" pivot, via an active spring element of variable length, and via a pair of Hill-type muscle models representing an extensor/flexor system. Due to mechanical feedback of passive elastic forces, the stable gaits of the conservative models are preserved, and now energy input and absorption balances to additionally stabilize a preferred speed, with only modest neural sensing and feedback being required. However, these bipedal models still cannot simultaneously match observed moment-yaw magnitudes and fore-aft dynamics.

Research paper thumbnail of Nonlinear Propagation of Light in One-Dimensional Periodic Structures

Research paper thumbnail of Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model

Research paper thumbnail of Mechanical models for insect locomotion: stability and parameter studies

Research paper thumbnail of The physics of optimal decision making: A formal analysis of models of performance in two alternative forced choice tasks

In this article, the authors consider optimal decision making in two-alternative forced-choice (T... more In this article, the authors consider optimal decision making in two-alternative forced-choice (TAFC) tasks. They begin by analyzing 6 models of TAFC decision making and show that all but one can be reduced to the drift diffusion model, implementing the statistically optimal algorithm (most accurate for a given speed or fastest for a given accuracy). They prove further that there is always an optimal trade-off between speed and accuracy that maximizes various reward functions, including reward rate (percentage of correct responses per unit time), as well as several other objective functions, including ones weighted for accuracy. They use these findings to address empirical data and make novel predictions about performance under optimality.

Research paper thumbnail of Globally Coupled Oscillator Networks

We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimens... more We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimensional tori. We focus on the efiects of symmetry and of the forms of the coupling functions, derived from un- derlying Hodgkin-Huxley type neuron models, on the existence, stability, and degeneracy of phase-locked solutions in which subgroups of oscillators share common phases. We also estimate domains of attraction for

Research paper thumbnail of Simple Neural Networks that Optimize Decisions

International Journal of Bifurcation and Chaos, 2005

We review simple connectionist and flring rate models for mutually inhibiting pools of neurons th... more We review simple connectionist and flring rate models for mutually inhibiting pools of neurons that discriminate between pairs of stimuli. Both are two-dimensional nonlinear stochastic ordinary difierential equations, and although they difier in how inputs and stim- uli enter, we show that they are equivalent under state variable and parameter coordinate changes. A key parameter is gain: the maximum slope

Research paper thumbnail of Optimality and some of its discontents: successes and shortcomings of existing models for binary decisions

Topics in cognitive science, 2014

We review how leaky competing accumulators (LCAs) can be used to model decision making in two-alt... more We review how leaky competing accumulators (LCAs) can be used to model decision making in two-alternative, forced-choice tasks, and we show how they reduce to drift diffusion (DD) processes in special cases. As continuum limits of the sequential probability ratio test, DD processes are optimal in producing decisions of specified accuracy in the shortest possible time. Furthermore, the DD model can be used to derive a speed-accuracy trade-off that optimizes reward rate for a restricted class of two alternative forced-choice decision tasks. We review findings that compare human performance with this benchmark, and we reveal both approximations to and deviations from optimality. We then discuss three potential sources of deviations from optimality at the psychological level--avoidance of errors, poor time estimation, and minimization of the cost of control--and review recent theoretical and empirical findings that address these possibilities. We also discuss the role of cognitive contr...

Research paper thumbnail of The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

Siam Review, 2006

Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy... more Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal

Research paper thumbnail of Lateralized Readiness Potentials Reveal Properties of a Neural Mechanism for Implementing a Decision Threshold

PLoS ONE, 2014

Many perceptual decision making models posit that participants accumulate noisy evidence over tim... more Many perceptual decision making models posit that participants accumulate noisy evidence over time to improve the accuracy of their decisions, and that in free response tasks, participants respond when the accumulated evidence reaches a decision threshold. Research on the neural correlates of these models' components focuses primarily on evidence accumulation. Far less attention has been paid to the neural correlates of decision thresholds, reflecting the final commitment to a decision. Inspired by a model of bistable neural activity that implements a decision threshold, we reinterpret human lateralized readiness potentials (LRPs) as reflecting the crossing of a decision threshold. Interestingly, this threshold crossing preserves signatures of a drift-diffusion process of evidence accumulation that feeds in to the threshold mechanism. We show that, as our model predicts, LRP amplitudes and growth rates recorded while participants performed a motion discrimination task correlate with individual differences in behaviorally-estimated prior beliefs, decision thresholds and evidence accumulation rates. As such LRPs provide a useful measure to test dynamical models of both evidence accumulation and decision commitment processes non-invasively.

Research paper thumbnail of A comparative study of drift diffusion and linear ballistic accumulator models in a reward maximization perceptual choice task

Frontiers in neuroscience, 2014

We present new findings that distinguish drift diffusion models (DDMs) from the linear ballistic ... more We present new findings that distinguish drift diffusion models (DDMs) from the linear ballistic accumulator (LBA) model as descriptions of human behavior in a two-alternative forced-choice reward maximization (Rmax) task. Previous comparisons have not considered Rmax tasks, and differences identified between the models' predictions have centered on practice effects. Unlike the parameter-free optimal performance curves of the pure DDM, the extended DDM and LBA predict families of curves depending on their additional parameters, and those of the LBA show significant differences from the DDMs, especially for poorly discriminable stimuli that incur high error rates. Moreover, fits to behavior reveal that the LBA and DDM provide different interpretations of behavior as stimulus discriminability increases. Trends for threshold setting (caution) in the DDMs are consistent between fits, while in the corresponding LBA fits, thresholds interact with distributions of starting points in a ...

Research paper thumbnail of Simple networks that optimize decisions

International Journal of Bifurcation and Chaos - IJBC, 2004

Research paper thumbnail of Heteroclinic cycles and modulated traveling waves in systems with O(2) symmetry

Research paper thumbnail of The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

Research paper thumbnail of Kuramoto–Sivashinsky Dynamics on the Center–Unstable Manifold

SIAM Journal on Applied Mathematics, 1989

Research paper thumbnail of A Simply Stabilized Running Model

SIAM Journal on Applied Dynamical Systems, 2003

Research paper thumbnail of Can Monkeys Choose Optimally When Faced with Noisy Stimuli and Unequal Rewards?

PLoS Computational Biology, 2009

Research paper thumbnail of Nonlinear Muscles, Passive Viscoelasticity and Body Taper Conspire To Create Neuromechanical Phase Lags in Anguilliform Swimmers

PLoS Computational Biology, 2008

Locomotion provides superb examples of cooperation among neuromuscular systems, environmental rea... more Locomotion provides superb examples of cooperation among neuromuscular systems, environmental reaction forces, and sensory feedback. As part of a program to understand the neuromechanics of locomotion, here we construct a model of anguilliform (eel-like) swimming in slender fishes. Building on a continuum mechanical representation of the body as an viscoelastic rod, actuated by a traveling wave of preferred curvature and subject to hydrodynamic reaction forces, we incorporate a new version of a calcium release and muscle force model, fitted to data from the lamprey Ichthyomyzon unicuspis, that interactively generates the curvature wave. We use the model to investigate the source of the difference in speeds observed between electromyographic waves of muscle activation and mechanical waves of body curvature, concluding that it is due to a combination of passive viscoelastic and geometric properties of the body and active muscle properties. Moreover, we find that nonlinear force dependence on muscle length and shortening velocity may reduce the work done by the swimming muscles in steady swimming.

Research paper thumbnail of Strong NLS soliton–defect interactions

Physica D: Nonlinear Phenomena, 2004