Intersegmental coordination in the lamprey: simulations using a network model without segmental boundaries

Swimming in vertebrates such as eel and lamprey involves the coordination of alternating left and right activity in each segment. Forward swimming is achieved by a lag between the onset of activity in consecutive segments rostrocaudally along the spinal cord. The intersegmental phase lag is approximately 1% of the cycle duration per segment and is independent of the swimming frequency. Since the lamprey has approximately 100 spinal segments, at any given time one wave of activity is propagated along the body. Most previous simulations of intersegmental coordination in the lamprey have treated the cord as a chain of coupled oscillators or well-defined segments. Here a network model without segmental boundaries is described which can produce coordinated activity with a phase lag. This ‘continuous’ pattern-generating network is composed of a column of 420 excitatory interneurons (E1 to E420) and 300 inhibitory interneurons (C1 to C300) on each half of the simulated spinal cord. The interneurons are distributed evenly along the simulated spinal cord, and their connectivity is chosen to reflect the behavior of the intact animal and what is known about the length and strength of the synaptic connections. For example, E100 connects to all interneurons between E51 and E149, but at varying synaptic strengths, while E101 connects to all interneurons between E52 and E150. This unsegmented E-C network generates a motor pattern that is sampled by output elements similar to motoneurons (M cells), which are arranged along the cell column so that they receive input from seven E and five C interneurons. The M cells thus represent the summed excitatory and inhibitory input at different points along the simulated spinal cord and can be regarded as representing the ventral root output to the myotomes along the spinal cord. E and C interneurons have five simulated compartments and Hodgkin-Huxley based dynamics. The simulated network produces rhythmic output over a wide range of frequencies (1–11 Hz) with a phase lag constant over most of the length, with the exception of the ‘cut’ ends due to reduced synaptic input. As the inhibitory C interneurons in the simulation have more extensive caudal than rostral projections, the output of the simulation has positive phase lags, as occurs in forward swimming. However, unlike the biological network, phase lags in the simulation increase significantly with burst frequency, from 0.5% to 2.3% over the range of frequencies of the simulation. Local rostral or caudal increases in excitatory drive in the simulated network are sufficient to produce motor patterns with increased or decreased phase lags, respectively. Received: 15 December 1995 / Accepted in revised form: 17 September 1996     

On the derivation and tuning of phase oscillator models for lamprey central pattern generators

Using phase response curves and averaging theory, we derive phase oscillator models for the lamprey central pattern generator from two biophysically-based segmental models. The first one relies on network dynamics within a segment to produce the rhythm, while the second contains bursting cells. We study intersegmental coordination and show that the former class of models shows more robust behavior over the animal's range of swimming frequencies. The network-based model can also easily produce approximately constant phase lags along the spinal cord, as observed experimentally. Precise control of phase lags in the network-based model is obtained by varying the relative strengths of its six different connection types with distance in a phase model with separate coupling functions for each connection type. The phase model also describes the effect of randomized connections, accurately predicting how quickly random network-based models approach the determinisitic model as the number of connections increases.     

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 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.     

Modeling of Substance P and 5-HT Induced Synaptic Plasticity in the Lamprey Spinal CPG: Consequences for Network Pattern Generation

Consequences of synaptic plasticity in the lamprey spinal CPG are analyzed by means of simulations. This is motivated by the effects substance P (a tachykinin) and serotonin (5-hydroxytryptamin; 5-HT) have on synaptic transmission in the locomotor network. Activity-dependent synaptic depression and potentiation have recently been shown experimentally using paired intracellular recordings. Although normally activity-dependent plasticity presumably does not contribute to the patterning of network activity, this changes in the presence of the neuromodulators substance P and 5-HT, which evoke significant plasticity. Substance P can induce a faster and larger depression of inhibitory connections but potentiation of excitatory inputs, whereas 5-HT induces facilitation of both inhibitory and excitatory inputs. Changes in the amplitude of the first postsynaptic potential are also seen. These changes could thus be a potential mechanism underlying the modulatory role these substances have on the rhythmic network activity.The aim of the present study has been to implement the activity dependent synaptic depression and facilitation induced by substance P and 5-HT into two alternative models of the lamprey spinal locomotor network, one relying on reciprocal inhibition for bursting and one in which each hemicord is capable of oscillations. The consequences of the plasticity of inhibitory and excitatory connections are then explored on the network level.In the intact spinal cord, tachykinins and 5-HT, which can be endogenously released, increase and decrease the frequency of the alternating left-right burst pattern, respectively. The frequency decreasing effect of 5-HT has previously been explained based on its conductance decreasing effect on K _Ca underlying the postspike afterhyperpolarization (AHP). The present simulations show that short-term synaptic plasticity may have strong effects on frequency regulation in the lamprey spinal CPG. In the network model relying on reciprocal inhibition, the observed effects substance P and 5-HT have on network behavior (i.e., a frequency increase and decrease respectively) can to a substantial part be explained by their effects on the total extent and time dynamics of synaptic depression and facilitation. The cellular effects of these substances will in the 5-HT case further contribute to its network effect.     

A model for intersegmental coordination in the leech nerve cord

The neuronal circuits that generate swimming movements in the leech were simulated by a chain of coupled harmonic oscillators. Our model incorporates a gradient of rostrocaudally decreasing cycle periods along the oscillator chain, a finite conduction delay for coupling signals, and multiple coupling channels connecting each pair of oscillators. The interactions mediated by these channels are characterized by sinusoidal phase response curves. Investigations of this model were carried out with the aid of a digital computer and the results of a variety of manipulations were compared with data from analogous physiological experiments. The simulations reproduced many aspects of intersegmental coordination in the leech, including the findings that: 1) phase lags between adjacent ganglia are larger near the caudal than the rostral end of the leech nerve cord; 2) intersegmental phase lags increase as the number of ganglia in nervecord preparations is reduced; 3) severing one of the paired lateral connective nerves can reverse the phase lag across the lesion and 4) blocking synaptic transmission in midganglia of the ventral nerve cord reduces phase lags across the block.     

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     

Spinal Mechanisms in the Control of Lamprey Swimming

SYNOPSIS. The lamprey, an anguilliform fish, swims using lateralundulatory movement; a transverse wave passes backward, fromhead to tail, the amplitude of the wave increasing as it movestailward. The wave of muscle activity producing this movementtravels down the body faster than the mechanical wave. The wayin which certain features of anguilliform movement contributeto its efficiency have been described. The neural activity underlyingswimming is characterized by: 1) rhythmical alternation betweenthe two sides of a single segment; 2) a burst duration thatremains a constant proportion of the cycle time and is independentof the cycle frequency; 3) rostrocaudal phase lag that is constantand also independent of the cycle frequency. Local circuitsin the lamprey spinal cord can generate this locomotory patternin the absence of sensory feedback following activation of excitatoryamino acid receptors; the pattern is centrally generated. Ithas been hypothesized that the spinal central pattern generatorfor locomotion consists of a series of segmental burst generatorscoupled together by an intersegmental coordinating system. Theintersegmental coordinating system functions to keep the frequenciesof the oscillators along the cord constant and to provide theappropriate rostrocaudal phase lag. Mechanosensitive units withinthe spinal cord are sensitive to movement of the spinal cord\notochordand movement of the spinal cord/notochord can entrain the burstpattern. Entrainment occurs through movement-related feedbackonto neurons at the local level. The possible roles this movement-relatedfeedback plays during locomotion are discussed.     

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.     

Entrainment ranges of forced phase oscillators

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest–neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest–neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.     

Neuronal Network Models of Phase Separation Between Limb CPGs of Digging Sand Crabs

Ordinary differential equations are used to model a peculiar motor behaviour in the anomuran decapod crustacean Emerita analoga. Little is known about the neural circuitry that permits E. analoga to control the phase relationships between movements of the fourth legs and pair of uropods as it digs into sand, so mathematical models might aid in identifying features of the neural structures involved. The geometric arrangement of segmental ganglia controlling the movements of each limb provides an intuitive framework for modelling. Specifically, due to the rhythmic nature of movement, the network controlling the fourth legs and uropods is viewed as three coupled identical oscillators, one dedicated to the control of each fourth leg and one for the pair of uropods, which always move in bilateral synchrony. Systems of Morris–Lecar equations describe the voltage and ion channel dynamics of neurons. Each central pattern generator for a limb is first modelled as a single neuron and then, more realistically as a multi-neuron oscillator. This process results in high-dimensional systems of equations that are difficult to analyse. In either case, reduction to phase equations by averaging yields a two-dimensional system of equations where variables describe only each oscillator’s phase along its limit cycle. The behaviour observed in the reduced equations approximates that of the original system. Results suggest that the phase response function in the two dimensional system, together with minimal input from asymmetric bilateral coupling parameters, is sufficient to account for the observed behaviour.     

Coherency and connectivity in oscillating neural networks: linear partialization analysis

 This paper studies the relation between the functional synaptic connections between two artificial neural networks and the correlation of their spiking activities. The model neurons had realistic non-oscillatory dynamic properties and the networks showed oscillatory behavior as a result of their internal synaptic connectivity. We found that both excitation and inhibition cause phase locking of the oscillating activities. When the two networks excite each other the oscillations synchronize with zero phase lag, whereas mutual inhibition between the networks resulted in an anti-phase (half period phase difference) synchronization. Correlations between the activities of the two networks can also be caused by correlated external inputs driving the systems (common input). Our analysis shows that when the networks exhibit oscillatory behavior and the rate of the common input is smaller than a characteristic network oscillator frequency, the cross-correlation functions between the activities of two systems still carry information about the mutual synaptic connectivity. This information can be retrieved with linear partialization, removing the influence of the common input. We further explored the network responses to periodic external input. We found that when the input is of a frequency smaller than a certain threshold, the network responds with bursts at the same frequency as the input. Above the threshold, the network responds with a fraction of the input frequency. This frequency threshold, characterizing the oscillatory properties of the network, is also found to determine the limit to which linear partialization works. Received: 20 October 1995 / Accepted in revised form: 20 May 1996     

The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model

We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey.A simplified mathematical model of a limit cycle oscillator is presented which consists of only a single dependent variable, the phase (t). By coupling N such oscillators together we are able to generate stable phase locked motions which correspond to traveling waves in the spinal cord, thus simulating fictive swimming. We are also able to generate irregular drifting motions which are compared to the experimental data obtained from cords with selective surgical lesions.     

Self-organizing effects of spontaneous neural activity on the development of spinal locomotor circuits in vertebrates

Presented in this paper is a neural network model that can be used to investigate the possible self-organizing mechanisms occurring during the early ontogeny of spinal neural circuits in the vertebrate motor system. The neural circuit is composed of multiple types of neurons which correspond to motorneurons, Renshaw cells and a hypothetical class of interneurons. While the connectivity of this circuit is genetically predetermined, the efficacies of these connections – the synaptic s trengths – evolve in accordance with activity-dependent mechanisms which are initiated by the intrinsic, autonomous activity present in the developing spinal cord. Using Oja's rule, a modified Hebbian learning scheme for adjusting the values of the connections, the network stably self-organizes developing, in the process, reciprocally activated motorneuron pools analogous to those which exist in vivo. Received: 30 December 1996 / Accepted in revised form: 20 June 1997     

Dendritic and Synaptic Effects in Systems of Coupled Cortical Oscillators

We explore the influence of synaptic location and form on the behavior of networks of coupled cortical oscillators. First, we develop a model of two coupled somatic oscillators that includes passive dendritic cables. Using a phase model approach, we show that the synchronous solution can change from a stable solution to an unstable one as the cable lengthens and the synaptic position moves further from the soma. We confirm this prediction using a system of coupled compartmental models. We also demonstrate that when the synchronous solution becomes unstable, a bifurcation occurs and a pair of asynchronous stable solutions appear, causing a phase lag between the cells in the system. Then using a variety of coupling functions and different synaptic positions, we show that distal connections and broad synaptic time courses encourage phase lags that can be reduced, eliminated, or enhanced by the presence of active currents in the dendrite. This mechanism may appear in neural systems where proximal connections could be used to encourage synchrony, and distal connections and broad synaptic time courses could be used to produce phase lags that can be modulated by active currents.     

Intersegmental variation of afferent pathways to giant interneurons of the earthworm,Lumbricus terrestris L.

Summary We have investigated the connectivity of four classes of mechanosensory afferents to giant interneurons in the earthwormLumbricus. Three of these classes of afferents change their specification for connection to medial giant (MGF) and lateral giant (LGF) fibers along the length of the animal. Near the caudal end, stimulation of touch, pressure and small tactile fibers generates excitatory post-synaptic potentials, epsp's, in the two LGF's but not in the MGF. Near the rostral end these afferents produce much smaller epsp's in the LGFs but produce large epsp's in the MGF. In the middle region of the animal an overlap region exists where both giant fibers receive approximately equal inputs from these afferents. The amplitude of these inputs are reduced compared to the maxima seen at either end. The fourth class of sensory afferents investigated, the stretch neurons, have no synaptic effect on the giant fibers anywhere in the nerve cord.These results explain at least part of the basis, in neuronal connectivity, for the differences in response to tactile stimulation of the head and tail segments previously characterized in terms of behavior and giant fiber impulse activity. In this system developmental mechanisms generating synaptic connectivity patterns have coded certain classes of homologous afferent neurons and interneurons to make different connections in different segments.Abbreviations MGF medial giant fiber - LGF lateral giant fiber - SN1 first segmental root - SN2 second segmental root - SN3 third segmental root - RIN giant interneuron     

Neural Control of Interappendage Phase During Locomotion

Interappendage phasing of crayfish swimmeret movements dependsupon a central nervous system network of command, oscillator,and coordinating neurons. The command neurons serve to set thegeneral excitation level in each of the segmental oscillators.The oscillator neurons in each hemi-ganglion generate the rhythmicalternations of powerstroke and returnstroke motor neuron activity.The coordinating neurons transmit the precise timing informationabout the state of one oscillator to other oscillators. Thisinformation can serve to advance or to delay the motor burstsdriven by the other oscillators. Which effect is observed dependsupon the arrival time of the coordinating neuron discharge withinthe cycle period of the modulated oscillator. This type of modulationleads to the prediction that a stable interappendage phase canresult from situations where there is not a fixed excitabilitygradient among the segmental oscillators. This prediction hasbeen verified using a cut command neuron preparation.     

Modelling inter-segmental coordination of neuronal oscillators: synaptic mechanisms for uni-directional coupling during swimming in Xenopus tadpoles

Locomotion requires longitudinal co-ordination. We have examined uni-directional synaptic coupling processes between two classes of neuronal network oscillators: autonomously active intrinsic oscillators, and potential oscillators that lack sufficient excitatory drive for autonomous activity. We model such oscillator networks in the bilaterally-symmetrical, Xenopus tadpole spinal cord circuits that co-ordinate swimming. Glutamate coupling EPSPs can entrain a second oscillator of lower frequency provided their strength is sufficient. Fast (AMPA) EPSPs advance spiking on each cycle, while slow (NMDA) EPSPs increase frequency over many cycles. EPSPs can also enable rhythmicity in potential oscillators and entrain them. IPSPs operate primarily on a cycle-by-cycle basis. They can advance or delay spiking to entrain a second intrinsic oscillator with higher, equal or lower frequency. Bilaterally symmetrical coupling connections operate twice per cycle: once in each half-cycle, on each side of the receiving oscillator. Excitatory and inhibitory coupling allow entrainment in complimentary areas of parameter space.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

A theory of hippocampal memory based on theta phase precession

 The neural dynamics of the hippocampal network for memory encoding of novel temporal sequences is proposed based on the theta rhythm modulated firing of place cells called theta phase precession. It is hypothesized that theta phase precession is generated at the entorhinal cortex by phase locking between local field theta oscillation and neural oscillators and that the hippocampal closed network with feedforward and backward projections employ theta phase precession to create selectivity in the associative connections needed for temporal sequence storage. Our analyses and computer experiments reveal that the phase precession generated by phase locking instantaneously endows stable phase relations among neural activities in the successively changing neural population. It is concluded that theta phase precession provides biologically plausible dynamics for the memory encoding of novel temporal sequences as episodic events. Received: 18 December 2002 / Accepted: 18 March 2003 / Published online: 20 May 2003 Correspondence to: Y. Yamaguchi (e-mail: yokoy@brain.riken.go.jp, Fax: +81-48-4676938) Acknowledgements. The author would like to express acknowledgement to Drs. McNaughton and Skaggs for their discussion and comment and to Dr. Amari for his continuous encouragement. Further thanks are given to Mr. Haga and Dr. Wu for their discussion and cooperation.