EP1762003A2 - Procede et dispositif pour reinitialisation previsible independante de la phase - Google Patents

Procede et dispositif pour reinitialisation previsible independante de la phase

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Publication number
EP1762003A2
EP1762003A2 EP05749319A EP05749319A EP1762003A2 EP 1762003 A2 EP1762003 A2 EP 1762003A2 EP 05749319 A EP05749319 A EP 05749319A EP 05749319 A EP05749319 A EP 05749319A EP 1762003 A2 EP1762003 A2 EP 1762003A2
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European Patent Office
Prior art keywords
phase
oscillator
oscillation
stimulus
control
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EP05749319A
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German (de)
English (en)
Inventor
Rodolfo R. Llinas
Viktor B. Inst. of Applied Physics Kazantsev
Vladimir I. Inst. of Applied Physics Nekorkin
Vladimir I. Makarenko
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New York University NYU
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New York University NYU
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/049—Temporal neural networks, e.g. delay elements, oscillating neurons or pulsed inputs
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/06—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons
    • G06N3/063—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons using electronic means
    • G06N3/065—Analogue means

Definitions

  • the present invention relates to control systems and methods, and particularly to control systems and methods that exhibit phase-independent predictable resetting.
  • BACKGROUND INFORMATION The olivo-cerebellar network is the key neuronal circuit in the brain for providing higher-level motor control in vertebrates.
  • the network is based on ocillatory dynamics of inferior olive (IO) neurons controlled by an inhibitory feedback loop with the cerebellar cortex.
  • IO inferior olive
  • the oscillations of IO neurons are organized in space and time in the form of oscillatory phase clusters.
  • the clusters provide a dynamical representation of arbitrary motor intention patterns that are further mapped to the motor execution system. Being supplied with sensory inputs, the olivo-cerebellar network is capable of rearranging the clusters in the process of movement execution.
  • the IO neurons produce quasi-sinusoidal oscillations with definite amplitude and frequency. Action potentials occur at the peaks of the oscillations and, hence, have precise timing properties.
  • the application of extracellular stimuli to the IO neurons does not cause changes in oscillation amplitude and frequency. Stimulation produces only a shift of the oscillation phase and hence a time shift of the corresponding action potential. Moreover, the resulting phase depends only on the strength of the stimulus and does not depend on the point at which the stimulus is applied.
  • phase of the IO oscillators can be quickly reset to a desired phase regardless of the history of phase evolution.
  • Motor control requires highly coordinated signals driving a large number of muscles. recent studies have shown that motor intention patterns to start an arbitrary movement are formed in the olivo-cerebellar functional network. See, Llinas, R. (1991) in Motor Control: Concepts and Issues, eds. Humphrey, D. R. & Freund, H. J. (Wiley, New York), pp. 223-242; Llinas, R. (2001) I of the Vortex: From neurons to self, The MIT Press; Welsh, J.P.
  • the motor intention patterns represent a set of action potentials inter-coordinated in space and time innervating a corresponding set of motor neurons.
  • the main information characteristic i.e. the main information-bearing control parameter, is the mutual phase relationship between sequences of action potentials innervating different muscles.
  • the coordination mechanism is provided by the sequence of oscillatory events in the olivo- cerebellar system.
  • the IO neurons are capable of appropriately reconfiguring their oscillations so as to provide the required phase cluster pattern.
  • the IO neurons have an internal syncrhonization mechanism.
  • the dendrite of an IO neuron forms gap junctions with about 50 neighboring cells providing local oscillation synchrony. See Llinas, R. & Yarom, Y. (1981) J. Physiol. Lond.
  • the reset phase of an IO neuron is defined only by the characteristics of the resetting stimulus and does not depend on the moment of time (i.e., initial phase) at which the stimulus is applied. See Leznik, E., Makarenko, V. & Llinas, R. (2002) J. Neurosci. 22, 2804-2815. In this sense, the reset is self-referential in that it ignores the "history" of the system evolution. This is a key property which makes the IO neuron oscillators extraordinarily flexible for processing motor commands and adapting to current conditions. Moreover, different IO neuron oscillators even when uncoupled and remotely located from each other can be quickly synchronized in phase upon receiving the same stimulus. FIGs.
  • FIG. 1A-1D show intracellular recordings of spontaneous IO neuron oscillations at 2 Hz interrupted by extracellular stimuli.
  • an extracellular stimulus delivered at the dorsal border of the IO nucleus generated a full action potential followed by a membrane hyperpolarization in nearby neurons.
  • FIG. 1 A after extracellular stimulation (marked with an arrowhead), the oscillations disappeared for about 750 msec (boxed area 10) and then resumed with a different phase approximately.
  • the membrane potential was approximately 60 mV.
  • FIG. IB intracellular recordings of spontaneous (dashed black trace) and stimulus- evoked (solid black trace) oscillations from the same cell are superimposed.
  • the average of six individual stimulus-evoked oscillations had the same frequency as that of the spontaneous oscillations.
  • the results were similar to those shown for a single stimulus, but the reset time was prolonged (data not shown).
  • the stimulus- evoked IO oscillations averaged over several trials had the same frequency and amplitude as spontaneous oscillations and differed only in a phase shift.
  • FIG. ID the average of six traces of stimulus evoked oscillations (solid trace) and the recording of spontaneous oscillations (dashed trace) are superimposed.
  • the stimulus-evoked oscillations in the average trace have the same frequency and amplitude as the spontaneous oscillations and differ only in the phase shift.
  • the phase reset effect in IO neurons has two basic features: (i) the resulting phase after stimulation is independent of the initial phase and can be controlled by the characteristics of the stimulus; and (ii) being stimulated by the same stimulus, different cells oscillating at different phases are reset to the same phase, i.e. synchronized.
  • the key electrical properties of IO neurons are described in Kazantsev, V.B., Nekorkin V.I., Makarenko, V.I. & Llinas, R. (2003) Procs. Natl. Acad. Sci. USA 100 (32), 13064-13068; Velarde, M.G., Nekorkin, V.I., Kazantsev, V.B., Makarenko, V.I. & Llinas, R. (2002) Neural Networks 15, 5-10.
  • Self-referential phase resetting is possible using a biologically-based model in accordance with a first aspect of the present invention.
  • the present invention provides a model based on inferior olive physiology which is capable of self-referential phase reset.
  • SPR mechanisms are described as are applications of the phase control strategy for artificial automatic control systems using SPR.
  • an electrical circuit that mimics the oscillatory and self-referential phase resetting properties of an inferior olive neuron is described, as is a control system comprising one or more of such circuits.
  • FIGs. 1A through ID show intracellular recordings of spontaneous IO neuron
  • FIG. 2A shows the phase space behavior of an exemplary embodiment of a model in accordance with the present invention.
  • FIGs. 2B and 2C show oscillations and spiking of the model under different conditions.
  • FIG. 3 A illustrates phase resetting in an exemplary oscillatory unit in accordance with the present invention.
  • FIGs. 3B and 3C illustrate phase resetting in multiple oscillatory units provided with the same excitatory and inhibitory stimulus, respectively.
  • FIG. 4A shows the deviation of the reset phase with respect to the amplitude of the resetting stimulus.
  • FIG. 4B shows how the reset phase varies with the amplitude of the resetting stimulus.
  • FIG. 5 A and 5B illustrate the phase space behavior of an exemplary oscillator in accordance with the present invention under excitatory and inhibitory stimulation, respectively.
  • FIGs. 6 A and 6B illustrate the formation of stimulus-induced patterns in a network of IO oscillatory units for excitatory and inhibitory stimuli, respectively.
  • FIG. 7 is a schematic representation of an exemplary embodiment of an oscillator circuit in accordance with the present invention.
  • FIG. 8 is a schematic representation of an exemplary embodiment of a control system in accordance with the present invention.
  • the present invention provides a model of individual IO neurons which reproduces their key electrical properties including self-referential phase resetting (SPR).
  • SPR self-referential phase resetting
  • the model of the present invention can be used to analyze the SPR effect, hi a further aspect of the present invention, the model is used to recreate the SPR effect for use in control systems, as described further below.
  • An exemplary embodiment of the model of the present invention comprises two coupled functional blocks. Oscillations emerge from supercritical Andronov-Hopf bifurcation in the first block which drive the dynamics of the second excitable block. When reaching the threshold at the peak of a subthreshold oscillation, the model generates a spike.
  • the timing of the spiking is thus determined by the sub-threshold oscillations.
  • the model qualitatively reproduces the spontaneous and stimuli-induced oscillations that have been observed in IO neurons.
  • An exemplary embodiment of a model in accordance with the present invention which exhibits the electrical behavior of IO neurons can be described by a set of four nonlinear differential equations as follows:
  • the variables z and w are responsible for the subthreshold oscillations and low- threshold (Ca2+-dependent) spiking, whereas the variables u and v describe the higher- threshold (Na +-dependent) spiking.
  • the parameters ec a and e ⁇ a control the oscillation time scales.
  • the parameters Ic a and I N3 drive the depolarization level of the two functional blocks.
  • the parameter a controls the shape of the nonlinear function /(x), particularly the excitation threshold, as well as the shape of the oscillator output during application of the excitation pulse.
  • the parameter k sets a relative time scale between the (u, v) and (z, w) blocks.
  • the function I ex t(t) describes the extracellural stimulus. It has a nonzero value, I ext (t)
  • I st and ⁇ s are constants describing the magnitude and duration, respectively, of the stimulus pulse arriving at the time instants t;.
  • the oscillations appear in the (z, w) subsystem with a frequency and amplitude that are controlled by the depolarization parameter Ic a -
  • the corresponding limit cycle in the (z, w) phase space is shown in FIG. 2A.
  • the nullclines are shown by the dashed curves.
  • the arrows illustrate fast and slow motions.
  • FIG. 2B shows subthreshold oscillations with Ca 2+ -dependent spiking. In this case, the (u, v) subsystem is unexcited.
  • FIG. 2C shows Na + -dependent spiking at the peaks of the subthreshold oscillation.
  • phase ⁇
  • R ⁇ z ' w
  • phase, ⁇ is a free parameter and can be set to an
  • the lower oscillation trace corresponds to the base oscillator, which is unperturbed.
  • the upper signal corresponds to the stimulus.
  • the stimulus is applied in-phase with the reference oscillator, i.e. it is applied as the reference
  • FIG. 3B shows the superimposed oscillations of 20 oscillators with different initial phases uniformly distributed over the range [0, 2 ⁇ ].
  • the 20 oscillators operate in accordance with the above-described model. After being reset by a stimulus pulse, the phases of the oscillators are localized in a narrow range with a mean phase ⁇ *. As FIG. 3B illustrates, the reset properties of the oscillators are independent on the initial state of the oscillator.
  • FIG. 3C shows similar behavior in the case of an inhibitory stimulus.
  • the first figure shows similar behavior in the case of an inhibitory stimulus.
  • the second figure shows similar behavior in the case of an inhibitory stimulus.
  • the values shown in FIG. 4A are based on 100
  • FIG. 4A shows the deviation in the case of stimulation with doublet pulses of duration 0.4T and having an inter-pulse interval of 12T . Note from FIG. 4A that for small stimulus amplitudes, the reset phases are distributed over the entire range [0, 2 ⁇ ] and the SPR effect disappears. In accordance with the SPR effect, the reset phase is controlled only by the characteristics of the stimulus, in which case the phase response curve representing the dependence of the reset phase on the initial phase is basically a constant line at the mean reset
  • FIG. 4B shows how the value of the mean reset phase ⁇ changes with increasing
  • the green arrows show how the desired value of the reset phase can be obtained by the appropriate choice of stimulus amplitude for excitatory, I st > 0, as well as inhibitory, I s t ⁇ 0, stimulation.
  • the curve of FIG. 4B covers the entire range [0, 2 ⁇ .
  • FIG. 4B shows that there is a point-by-point correspondence between the stimulus amplitude and the reset phase and that any desired value of the oscillation phase can be set by appropriate choice of the stimulus amplitude. Consequently, at any moment of time, one can control the oscillation phase independently of the "history" of the system evolution. Note that the SPR effect also takes place for l s t 0, i.e. for inhibitory extracellular stimuli. This is also shown in FIGs. 3C and 4A.
  • FIG. 5 A illustrates how multiple points (e.g., 100) uniformly distributed along the limit cycle are transformed under an excitatory stimulus.
  • the phase volume occupied by the limit cycle becomes strongly compressed, converging to the reset point after the stimulation.
  • FIG. 5A shows a sequence of snapshots of the transformation of the initial limit cycle under an excitatory stimulus.
  • the initial circle is compressed while evolving along the right and left compartment of the slow motion manifold located near the z-nullcline (dashed curve).
  • the slow motion manifold refers to an isolated curve on the phase plane which is invariant relative to the system trajectories.
  • FIG. 5B illustrates the phase volume transformation for an inhibitory stimulus, hi this case, the reset occurs faster (the excursion of the points is shorter) but with less precision. In this case the circle is compressed only near the left compartment of the slow motions manifold.
  • the w-nullcline When a sufficiently strong stimulus is applied, the w-nullcline is shifted either to the right part of the nonlinear curve (excitatory stimulus) or to the left part (inhibitory stimulus). Then, while a stimulus is applied, the middle part becomes unstable and the trajectories leave it to the region of fast motion. Note that due to different time scales, the initial circle becomes strongly compressed. This compression can be treated with Lyapunov exponents indicating exponential changing of a phase volume along trajectories.
  • ⁇ can be estimated using Equation Set 1 to be: ⁇ f(zo)/e, [3] where z 0 is the coordinate of the points at the manifold.
  • z 0 is the coordinate of the points at the manifold.
  • the Lyapunov exponent is strongly negative for small e. This corresponds to the strong phase volume compression in the transverse direction when the trajectories evolve near the manifold.
  • the volume becomes elongated near the manifolds.
  • the trajectories jump once more into the fast motion region elongating in the horizontal direction.
  • the phase volume again compresses near the left stable compartment of the manifold (FIG. 5A).
  • phase volume the limit cycle circle
  • the condition of the reset is I st ⁇ z m j n - Ic that provides the stability of the fixed point during the stimulation.
  • phase volume compression is provided by the Lyapunov eigenvalues of the stable fixed point that appears due to the stimulus.
  • SPR-induced synchronization Applying the SPR effect to large ensembles of oscillatory units can provide their phase synchronization. If a large number of isolated IO units are stimulated by the same pulse, they will return the same phase and hence become mutually synchronized. (See FIGs. 3B and 3C).
  • SPR-induced synchronization does not require coupling among the oscillators. Even remotely located cells therefore can be phase synchronized by just one or a few stimulation pulses.
  • the oscillators in the network can be reset to oscillate at any relative phase, as shown in FIG. 4B.
  • stimulus parameters i.e., amplitude and duration
  • the oscillators in the network can be reset to oscillate at any relative phase, as shown in FIG. 4B.
  • one or more oscillators receiving a first stimulus can be reset to oscillate with any desired phase relative to one or more other oscillators.
  • Different groups or clusters of oscillators (each group including at least one oscillator) can be stimulated with pulses of different amplitude and/or duration to yield different relative phases among the groups.
  • phase clusters of any complex spatial configuration can be formed.
  • the cluster configuration can be easily rearranged irrespective of the cluster configuration that may have existed before the stimulus. This property is realized in the olivo-cerebellar network for implementing motor intention patterns.
  • FIGs. 6A and 6B SPR-induced synchronization as implemented with an exemplary embodiment of a universal control system in accordance with the present invention is illustrated in FIGs. 6A and 6B.
  • a square network of 200 x 200 locally coupled IO oscillators each implemented in accordance with the above-described exemplary model, is stimulated by a given input pattern.
  • the pulse amplitudes vary over a range I st e [1 st 1 , 1 st 2 ] •
  • the units are uncoupled.
  • the initial phases are randomly distributed over the range [0, 2 ⁇ ].
  • the stimulus amplitude pattern is taken from a digitized grayscale 200x200 image,
  • the image, c[i,j], contains a picture of a bug.
  • each unit Upon stimulation, each unit will be reset to a certain phase, as determined in accordance with the relationship shown in FIG. 4B.
  • the stimulus pattern is the grayscale image of a bug digitized and mapped into the range of 1 9 amplitudes [Ist , Ist ].
  • the network evolves to a phase distribution 602 corresponding to the bug image.
  • a second stimulus pattern of the same image is applied and the system reproduces the desired phase distibution 603 corresponding to the bug image.
  • the image appears inverted due to the negative slope of the amplitude-phase curve (see FIG. 4B). Since inhibitory resetting is less precise, some details of the picture are missed after the first stimulus.
  • the second stimulus corrects the misprints reproducing the inverted bug picture.
  • the images are plotted with the same grayscale grade of the phase
  • the amplitude-phase curve of FIG. 4B is piece-wise linear. Therefore, nonlinear image distortions during the transformation are negligible.
  • the phase distribution in the network is further translated to corresponding action potential patterns when the IO units fire Na + -dependent spikes (i.e., as represented by the u, v variables in Equation Set 1 and shown in FIG. 2C) at the peak times of the oscillations.
  • the units When stimulated, the units are effectively uncoupled due to the inhibitory feedback.
  • the system tends to sustain the cluster configuration if no other stimuli are applied.
  • In-phase oscillators have a shorter coupling inhibition, hence become effectively coupled and thereby sustaining the synchronization.
  • Out-of-phase oscillators have a prolonged coupling inhibition because the inhibition periods are summed from the two units. They therefore stay effectively uncoupled, sustaining their phase difference.
  • the example of the SPR effect in the UCS-based network represents, in fact, the mechanism of sensory-motor transformation in the brain.
  • the sensorial information in the form of sequences of action potentials appropriately reset the motor control oscillators to a phase pattern that is further converted to a space-time distribution of action potentials for implementing motor execution patterns.
  • the SPR effect experimentally observed in IO neurons can be effectively modelled using the nonlinear dynamical model of the present invention.
  • the SPR property of IO neurons plays a role in the global functions of the olivo-cerebellar network for providing motor control.
  • the system is extraordinarily flexible to implement a given motor intention and to modify it in real time according to sensorial information.
  • the system does not need to continuously keep its current configuration, being able to successively reset to a given pattern from any state, hi other words, the system does not need any operation memory, making it very reliable and preventing "computational overloads" that occur when memorizing the states.
  • the SPR effect is quite fast (on the order of an oscillation period), allowing the system to operate in real time.
  • the SPR property of IO oscillators in accordance with the present invention allows them to be used advantageously in artificial control systems. Indeed, the IO oscillator represents a phase controller. One can set and keep a required phase by stimulating the oscillator with the appropriate pulse stimulus.
  • phase controller can maintain it at a desired level.
  • a physical parameter e.g., position, velocity, angle, temperature, etc.
  • the controlling principle here deals with "stumbling response.” See e.g., Yamasaki, T., Nomura, T & Sato, S. (2003) BioSystems.
  • a walking animal does not fall but resets its walking rhythm, irrespective of the moment at which the obstacle has appeared.
  • Another interesting SPR application concerns the synchronization of oscillatory systems. Multiple oscillators can be synchronized by stimulating them with the same stimulus pulse.
  • Such synchronization could be very powerful when the task is to synchronize a large array of oscillators or to synchronize spatially distant cells.
  • Such arrays (like the inferior olive) do not need a complex network of interconnections among the cells. Rather, a single stimulation signal generated by a master cell can be used to stimulate multiple cells at the same instant of time.
  • the SPR effect can be used to provide a mechanism whereby information can be represented and stored in the form of oscillatory clusters.
  • the fo ⁇ nation of oscillatory clusters can be extremely fast as one can speed up the oscillators up to the limits of the constituent materials. See Hopfield, J. J. (1982). PBAS 79, 2554-2558.
  • the SPR-based pattern formation can work directly with digitized information converted to the stimulus template. See Abott, L.F. (1990) J. Phys. A 23, 3835.
  • the exemplary circuit 700 shown can operate in a variety of modes. In a first mode, the circuit oscillates with a quasi-sinusoidal signal at approximately 930 Hz when power is applied at terminals 3 and 10, as shown, and terminals 2 and 6 are connected together. The oscillatory output can be monitored on terminal 2. In a second mode, the circuit 700 can operate with impulse stimulation. In this mode, terminals 1 and 6 are connected together. A pulse generator 750 is connected across terminals 1 and 2 to provide the pulse stimulation to the circuit. The output of the circuit 700 can be monitored at terminal 2.
  • +10 v is applied to terminal 3 and -10 v is applied to terminal 10 for power and the amplitude of the pulses generated by the pulse generator is 0.1 to 5.0 v, with a pulse duration of 1 msec.
  • the circuit 700 can be implemented with discrete components or in an integrated circuit.
  • the op-amps can be implemented, for example, using an MCI 458 op-amp circuit.
  • the component values shown are exemplary and can vary depending on the application.
  • FIG. 8 shows a block diagram of an exemplary controller 800 implemented with a plurality of oscillator circuits 810-870, each of which can be implemented using the exemplary circuit 700 described above.
  • the labels “1" and “2" indicate the terminals 1 and 2 of the exemplary circuit 700.
  • the oscillator 870 is set up to operate as a base oscillator, oscillating at a stable frequency (e.g., 930 Hz) and phase.
  • the amplitude of each pulse is 5 volts and the duration is less than 2 msec.
  • the pulses generated by the pulse generator 890 are provided to each oscillator 810- 860 via a pulse amplitude adjuster 812-862, respectively.
  • Each pulse amplitude adjuster 812- 862 individually adjusts the amplitude of the pulses applied to its respective oscillator 810- 860.
  • the pulse amplitude adjusters 812-862 comprise circuitry that may be capable of attenuating the pulses, amplifying the pulses, or both, h an exemplary embodiment, the pulse amplitude adjusters 812-862 comprise variable resistors arranged as variable resistor dividers to attenuate the pulses applied to their respective oscillators.
  • each oscillator 810-860 is provided to a respective phase detector 815- 865.
  • the output of the base oscillator 890 is provided to all of the phase detectors 815-865.
  • Each phase detector 815-865 generates a DC signal whose level is indicative of the phase difference between the corresponding oscillator 810-860 and the base oscillator 890.
  • the signals generated by the phase detectors can be used to control electro-mechanical devices or the like.

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Abstract

Aux fins de l'invention, la dynamique oscillatoire des neurones de l'olive inférieure (10) du réseau olivo-cérébellaire du cerveau est reproduite et appliquée à différentes applications de contrôle. On décrit un modèle neuronal de l'olive inférieure qui produit des oscillations quasi-sinusoïdales avec une amplitude et une fréquence caractéristiques. Les potentiels d'action interviennent aux crêtes des oscillations et ont des propriétés de synchronisation précises. Des stimuli appliqués à ce modèle ne font pas apparaître de changements dans l'amplitude et la fréquence d'oscillation, mais ils produisent un décalage de phase d'oscillation, et donc un décalage de synchronisation des potentiels d'action correspondants. La phase résultante est définie par l'intensité du stimulus et ne dépend pas du moment d'application, donnant ainsi un effet de réinitialisation de phase à auto-référence. Les oscillations de plusieurs neurones de l'olive inférieure, ou de leurs modèles, couplés ensemble, tendent à s'organiser dans l'espace et le temps sous la forme de groupes de phases oscillatoires. Les phases des oscillateurs de l'olive inférieure peuvent être rapidement réglées sur un profil de phase spécifique, indépendamment de l'historique d'évolution de phase. On décrit aussi un circuit de mise en oeuvre de modèle neuronal de l'olive inférieure et un système de contrôle reposant sur ce circuit.
EP05749319A 2004-05-05 2005-05-05 Procede et dispositif pour reinitialisation previsible independante de la phase Withdrawn EP1762003A2 (fr)

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