Activity-dependent modulation of adaptation produces a constant burst proportion in a model of the lamprey spinal locomotor generator

The neuronal network underlying lamprey swimming has stimulated extensive modelling on different levels of abstraction. The lamprey swims with a burst frequency ranging from 0.3 to 8–10 Hz with a rostro-caudal lag between bursts in each segment along the spinal cord. The swimming motor pattern is characterized by a burst proportion that is independent of burst frequency and lasts around 30%–40% of the cycle duration. This also applies in preparations in which the reciprocal inhibition in the spinal cord between the left and right side is blocked. A network of coupled excitatory neurons producing hemisegmental oscillations may form the basis of the lamprey central pattern generator (CPG). Here we explored how such networks, in principle, could produce a large frequency range with a constant burst proportion. The computer simulations of the lamprey CPG use simplified, graded output units that could represent populations of neurons and that exhibit adaptation. We investigated the effect of an active modulation of the degree of adaptation of the CPG units to accomplish a constant burst proportion over the whole frequency range when, in addition, each hemisegment is assumed to be self-oscillatory. The degree of adaptation is increased with the degree of stimulation of the network. This will make the bursts terminate earlier at higher burst rates, allowing for a constant burst proportion. Without modulated adaptation the network operates in a limited range of swimming frequencies due to a progressive increase of burst duration with increasing background stimulation. By introducing a modulation of the adaptation, a broad burst frequency range can be produced. The reciprocal inhibition is thus not the primary burst terminating factor, as in many CPG models, and it is mainly responsible for producing alternation between the left and right sides. The results are compared with the Morris-Lecar oscillator model with parameters set to produce a type A and type B oscillator, in which the burst durations stay constant or increase, respectively, when the background stimulation is increased. Here as well, burst duration can be controlled by modulation of the slow variable in a similar way as above. When oscillatory hemisegmental networks are coupled together in a chain a phase lag is produced. The production of a phase lag in chains of such oscillators is compared with chains of Morris-Lecar relaxation oscillators. Models relating to the intact versus isolated spinal cord preparation are discussed, as well as the role of descending inhibition. Received: 1 April 1997 / Accepted in revised form: 20 March 1998     

Energy efficient walking with central pattern generators: from passive dynamic walking to biologically inspired control

Like human walking, passive dynamic walking—i.e. walking down a slope with no actuation except gravity—is energy efficient by exploiting the natural dynamics. In the animal world, neural oscillators termed central pattern generators (CPGs) provide the basic rhythm for muscular activity in locomotion. We present a CPG model, which automatically tunes into the resonance frequency of the passive dynamics of a bipedal walker, i.e. the CPG model exhibits resonance tuning behavior. Each leg is coupled to its own CPG, controlling the hip moment of force. Resonance tuning above the endogenous frequency of the CPG—i.e. the CPG’s eigenfrequency—is achieved by feedback of both limb angles to their corresponding CPG, while integration of the limb angles provides resonance tuning at and below the endogenous frequency of the CPG. Feedback of the angular velocity of both limbs to their corresponding CPG compensates for the time delay in the loop coupling each limb to its CPG. The resonance tuning behavior of the CPG model allows the gait velocity to be controlled by a single parameter, while retaining the energy efficiency of passive dynamic walking.     

Estimating the Strength and Direction of Functional Coupling in the Lamprey Spinal Cord

A method of estimating coupling strength between two neural oscillators based on their spikes trains (Kiemel and Cohen, J. Comput. Neurosci. 5: 267–284, 1998) is tested using simulated data and then applied to experimental data from the central pattern generator (CPG) for swimming in the lamprey. The method is tested using a model of two connectionist oscillators and a model of two endogenously bursting cells. For both models, the method provides useful estimates of the relative strength of coupling in each direction, as well as estimates of total strength. The method is applied to pairs of motor-nerve recordings from isolated 50-segment pieces of spinal cords from adult silver lampreys (Ichthyomyzon unicuspus). The strength and direction of coupling is estimated under control conditions and conditions in which intersegmental coupling between the two recording locations is weakened by hemisections of the spinal cords and/or chambers containing an inhibitory solution that blocks firing in postsynaptic cells. The relevance of these measures in constraining models of the CPG is discussed.     

Mathematical models for the swimming pattern of a lamprey

A significant characteristic in a swimming pattern of a lamprey is the generation of a constant phase lag along its body in spite of the wide range of undulation frequencies. In this paper, we discuss a mathematical treatment for coupled oscillators with time-delayed interaction and propose a model for the central pattern generator (CPG) of a lamprey to account for the generation of a constant phase relation, with consideration of the signal conduction time. From this model, it is suggested that the desired phase relation can be produced by long ascending connections from the tail to the neck region of the CPG.     

The Calculation of Frequency-Shift Functions for Chains of Coupled Oscillators, with Application to a Network Model of the Lamprey Locomotor Pattern Generator

Chains of coupled limit-cycle oscillators are considered, in which the coupling is assumed to be weak and only between adjacent oscillators. For such a system the change in frequency of an oscillator due to the coupling can be expressed, up to first order in thecoupling strength, by functions that depend only on the phase difference between the coupled oscillators. In this article a numerical algorithm is developed for the evaluation of these functions (the H-functions) in terms of a single oscillator and the interactions between coupled oscillators. The technique is applied to a connectionist model for the locomotor pattern generator in the lamprey spinal cord.An H-function so derived is compared to a function derived empirically(the C-function) from simulations of the same system. The phase lagsthat develop between adjacent oscillators in a simulated chain are compared with those predicted theoretically, and it is shown that coupling thatis functionally strong is nonetheless weak enough to behave as predicted.     

Coupling of posture and gait: mode locking and parametric excitation

 We investigate the temporal coordination of human gait and posture and infer the nature of their coupling. Participants viewed a sinusoidally oscillating visual display which induced medial-lateral postural sway during treadmill walking, while display frequency was varied (0.075–1.025 Hz). First, postural responses exhibited the usual low-pass characteristic but with an additional resonance peak near the preferred stride frequency, although shifted downward by 0.12 Hz; this provides evidence of a coupling from gait to posture. Second, the step cycle adapted to mode lock with the visual driver and postural sway, as well as displaying instances of intermittency (slipping in and out of phase) and quasiperiodicity (phase wandering); this provides evidence of a coupling from posture to gait. We observed a spectrum of integer mode locks, including a large 1:1 trapping region about the stride frequency and superharmonic entrainment (stride frequency > driver frequency) at lower driver frequencies. A coupled-oscillator model that incorporates a novel parametric coupling from posture to the gait “stiffness” term reproduces these features of the data, including the resonance peak shift. Biological coordination patterns may thus emerge naturally as properties of a system of appropriately coupled oscillators. Received: 23 June 1999 / Accepted in revised form: 10 January 2001     

Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

We describe and analyze a model for a stochastic pulse-coupled neuronal network with many sources of randomness: random external input, potential synaptic failure, and random connectivity topologies. We show that different classes of network topologies give rise to qualitatively different types of synchrony: uniform (Erdős–Rényi) and “small-world” networks give rise to synchronization phenomena similar to that in “all-to-all” networks (in which there is a sharp onset of synchrony as coupling is increased); in contrast, in “scale-free” networks the dependence of synchrony on coupling strength is smoother. Moreover, we show that in the uniform and small-world cases, the fine details of the network are not important in determining the synchronization properties; this depends only on the mean connectivity. In contrast, for scale-free networks, the dynamics are significantly affected by the fine details of the network; in particular, they are significantly affected by the local neighborhoods of the “hubs” in the network.     

Analysis of clustered firing patterns in synaptically coupled networks of oscillators

Oscillators in networks may display a variety of activity patterns. This paper presents a geometric singular perturbation analysis of clustering, or alternate firing of synchronized subgroups, among synaptically coupled oscillators. We consider oscillators in two types of networks: mutually coupled, with all-to-all inhibitory connections, and globally inhibitory, with one excitatory and one inhibitory population of oscillators, each of arbitrary size. Our analysis yields existence and stability conditions for clustered states, along with formulas for the periods of such firing patterns. By using two different approaches, we derive complementary conditions, the first set stated in terms of time lengths determined by intrinsic and synaptic properties of the oscillators and their coupling and the second set stated in terms of model parameters and phase space structures directly linked to parameters. These results suggest how biological components may interact to produce the spindle sleep rhythm in thalamocortical networks. Received: 9 September 1999 / Revised version: 7 July 2000 / Published online: 24 November 2000     

Binding and segmentation of multiple objects through neural oscillators inhibited by contour information

 Temporal correlation of neuronal activity has been suggested as a criterion for multiple object recognition. In this work, a two-dimensional network of simplified Wilson-Cowan oscillators is used to manage the binding and segmentation problem of a visual scene according to the connectedness Gestalt criterion. Binding is achieved via original coupling terms that link excitatory units to both excitatory and inhibitory units of adjacent neurons. These local coupling terms are time independent, i.e., they do not require Hebbian learning during the simulations. Segmentation is realized by a two-layer processing of the visual image. The first layer extracts all object contours from the image by means of “retinal cells” with an “on-center” receptive field. Information on contour is used to selectively inhibit Wilson-Cowan oscillators in the second layer, thus realizing a strong separation among neurons in different objects. Accidental synchronism between oscillations in different objects is prevented with the use of a global inhibitor, i.e., a global neuron that computes the overall activity in the Wilson-Cowan network and sends back an inhibitory signal. Simulations performed in a 50×50 neural grid with 21 different visual scenes (containing up to eight objects + background) with random initial conditions demonstrate that the network can correctly segment objects in almost 100% of cases using a single set of parameters, i.e., without the need to adjust parameters from one visual scene to the next. The network is robust with reference to dynamical noise superimposed on oscillatory neurons. Moreover, the network can segment both black objects on white background and vice versa and is able to deal with the problem of “fragmentation.” The main limitation of the network is its sensitivity to static noise superimposed on the objects. Overcoming this problem requires implementation of more robust mechanisms for contour enhancement in the first layer in agreement with mechanisms actually realized in the visual cortex. Received: 25 October 2001 / Accepted: 26 February 2003 / Published online: 20 May 2003 Correspondence to: Mauro Ursino (e-mail: mursino@deis.unibo.it, Tel.: +39-051-2093008, Fax: +39-051-2093073)     

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.     

On partial contraction analysis for coupled nonlinear oscillators

We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in flocks of oscillators or dynamic elements.     

Intersegmental coordinating system of the lamprey central pattern generator for locomotion

Summary In the lamprey,Ichthyomyzon unicuspis, the wave of activity required for normal swimming movements can be generated by a central pattern generator (CPG) residing in the spinal cord. A constant phase coupling between spinal segments can be organized by intersegmental coordinating neurons intrinsic to the cord. The rostral and caudal segmental oscillators of the CPG have different preferred frequencies when separated from each other. Therefore the system must maintain the segmental oscillators of the locomotor CPG at a single common frequency and with the proper relative timing. Using selective lesions and a split-bath, it is demonstrated that the coordinating system is comprised of at least 3 subsystems, short-axon systems in the lateral and medial tracts and a long axon system in the lateral tracts. Each alone can sustain relatively stable coordinated activity.Abbreviations CPG central pattern generator - NMDA N-methyl-D-aspartate - VR ventral root     

The Jackprot Simulation Couples Mutation Rate with Natural Selection to Illustrate How Protein Evolution Is Not Random

Protein evolution is not a random process. Views which attribute randomness to molecular change, deleterious nature to single-gene mutations, insufficient geological time, or population size for molecular improvements to occur, or invoke “design creationism” to account for complexity in molecular structures and biological processes, are unfounded. Scientific evidence suggests that natural selection tinkers with molecular improvements by retaining adaptive peptide sequence. We used slot-machine probabilities and ion channels to show biological directionality on molecular change. Because ion channels reside in the lipid bilayer of cell membranes, their residue location must be in balance with the membrane’s hydrophobic/philic nature; a selective “pore” for ion passage is located within the hydrophobic region. We contrasted the random generation of DNA sequence for KcsA, a bacterial two-transmembrane-domain (2TM) potassium channel, from Streptomyces lividans, with an under-selection scenario, the “jackprot,” which predicted much faster evolution than by chance. We wrote a computer program in JAVA APPLET version 1.0 and designed an online interface, The Jackprot Simulation , to model a numerical interaction between mutation rate and natural selection during a scenario of polypeptide evolution. Winning the “jackprot,” or highest-fitness complete-peptide sequence, required cumulative smaller “wins” (rewarded by selection) at the first, second, and third positions in each of the 161 KcsA codons (“jackdons” that led to “jackacids” that led to the “jackprot”). The “jackprot” is a didactic tool to demonstrate how mutation rate coupled with natural selection suffices to explain the evolution of specialized proteins, such as the complex six-transmembrane (6TM) domain potassium, sodium, or calcium channels. Ancestral DNA sequences coding for 2TM-like proteins underwent nucleotide “edition” and gene duplications to generate the 6TMs. Ion channels are essential to the physiology of neurons, ganglia, and brains, and were crucial to the evolutionary advent of consciousness. The Jackprot Simulation illustrates in a computer model that evolution is not and cannot be a random process as conceived by design creationists.     

Pyramid power: Principal cells of the hippocampus unite!

Electrical transmission in the mammalian brain is now well established. A new study by Thomson and colleagues elegantly demonstrates coupling between CA1 hippocampal pyramidal cells, which is far more common than previously supposed. Although the history of coupling is extensive, doubt, predjudice, and technical issues long kept it from wide acceptance. Here “spikelets” or “fast prepotentials” are found when two cells are coupled and in this situation result from electrical transmission of impulses from one coupled cell to the other. Interesting questions remain as to whether connexin or pannexin gap junctions serve as the molecular substrate of transmission, and the role of electrical transmission in hippocampal physiology is uncertain. Increased coupling could well contribute to the known tendency of the hippocampus to exhibit seizure activity.     

Neural control of interlimb oscillations

How do humans and other animals accomplish coordinated movements? How are novel combinations of limb joints rapidly assembled into new behavioral units that move together in in-phase or anti-phase movement patterns during complex movement tasks? A neural central pattern generator (CPG) model simulates data from human bimanual coordination tasks. As in the data, anti-phase oscillations at low frequencies switch to in-phase oscillations at high frequencies, in-phase oscillations occur at both low and high frequencies, phase fluctuations occur at the anti-phase in-phase transition, a “seagull effect” of larger errors occurs at intermediate phases, and oscillations slip toward in-phase and anti-phase when driven at intermediate phases. These oscillations and bifurcations are emergent properties of the CPG model in response to volitional inputs. The CPG model is a version of the Ellias-Grossberg oscillator. Its neurons obey Hodgkin-Huxley type equations whose excitatory signals operate on a faster time scale than their inhibitory signals in a recurrent on-center off-surround anatomy. When an equal command or GO signal activates both model channels, the model CPG can generate both in-phase and anti-phase oscillations at different GO amplitudes. Phase transitions from either in-phase to anti-phase oscillations, or from anti-phase to in-phase oscillations, can occur in different parameter ranges, as the GO signal increases. Received: 22 August 1994 / Accepted in revised form: 13 May 1997     

Intersegmental phase lags in the lamprey spinal cord: Experimental confirmation of the existence of a boundary region

A critical feature of the motor pattern generated by the lamprey spinal cord is an intersegmental delay that is constant down the cord and scales with cycle duration. This has been modelled as the output of a chain of coupled oscillators, within a general mathematical framework developed by Kopell and Ermentrout (1986, 1988). The analysis predicts that for asymmetric coupling of equally-activated oscillators, the intersegmental phase lag will be uniform along the chain except in a boundary layer at one end. Here we provide experimental evidence that a boundary layer does occur at the rostral end of an isolated preparation of lamprey spinal cord. In the context of the mathematical analysis, this indicates that ascending coupling is dominant in the control of intersegmental phase lag in the lamprey.     

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.     

Some selection models in catalytically coupled selfreproducing macromolecular systems

Some models for selection in enzyme coupled, selfreproducing, macromolecular systems are considered. The enzyme coupling provides a mechanism for the buildup of systems with large information content. The concept of “selective value” and criteria for selection is discussed.     

Spatial and temporal stochastic interaction in neuronal assemblies

Summary The observation of various types of spatio-temporal correlations in spike-patterns of multiple cortical neurons has shifted attention from rate coding paradigms to computational processes based on the precise timing of spikes in neuronal ensembles. In the present work we develop the notion of “spatial” and “temporal interaction” which provides measures for statistical dependences in coupled stochastic processes like multiple unit spike trains. We show that the classical Willshaw network and Abeles’ synfire chain model both reveal a moderate spatial interaction, but only the synfire chain model reveals a positive temporal interaction, too. Systems that maximize temporal interaction are shown to be almost deterministic globally, but posses almost unpredictable firing behavior on the single unit level.