A generalized locomotion CPG architecture based on oscillatory ...

Biol. Cybern. 89, 34–42 (2003) DOI 10.1007/s00422-003-0409-7 Ó Springer-Verlag 2003

A generalized locomotion CPG architecture based on oscillatory building blocks Zhijun Yang1 , Felipe M. G. Franc¸a2 1

School of Mathematics and Computer Science, Nanjing Normal University, Nanjing, 210097, China COPPE – Systems Engineering and Computer Science Program, Federal University of Rio de Janeiro, Caixa Postal 68511, Rio de Janeiro, RJ, 21941-972, Brazil

2

Received: 14 June 2002 / Accepted: 18 February 2003 / Published online: 20 May 2003

Abstract. Neural oscillation is one of the most extensively investigated topics of artificial neural networks. Scientific approaches to the functionalities of both natural and artificial intelligences are strongly related to mechanisms underlying oscillatory activities. This paper concerns itself with the assumption of the existence of central pattern generators (CPGs), which are the plausible neural architectures with oscillatory capabilities, and presents a discrete and generalized approach to the functionality of locomotor CPGs of legged animals. Based on scheduling by multiple edge reversal (SMER), a primitive and deterministic distributed algorithm, it is shown how oscillatory building block (OBB) modules can be created and, hence, how OBB-based networks can be formulated as asymmetric Hopfield-like neural networks for the generation of complex coordinated rhythmic patterns observed among pairs of biological motor neurons working during different gait patterns. It is also shown that the resulting Hopfield-like network possesses the property of reproducing the whole spectrum of different gaits intrinsic to the target locomotor CPGs. Although the new approach is not restricted to the understanding of the neurolocomotor system of any particular animal, hexapodal and quadrupedal gait patterns are chosen as illustrations given the wide interest expressed by the ongoing research in the area.

1 Introduction For a long time the possibility of the existence of central pattern generators (CPGs) and their possible role in biological rhythmic activities such as locomotion have inspired great interest in the scientific community. As many neurophysiologists are still arguing and tending to agree on the existence of some forms of CPGs, several mathematically strict models of CPGs and their effects Correspondence to: Z. Yang (e-mail: [email protected])

on human and animal locomotion (Williams et al. 1990; Golubitsky et al. 1998, 1999) have been proposed. Some therapeutists have also assumed this plausible mechanism for rehabilitation purposes (Hess et al. 1994). This paper proposes a generalized discrete model for production of a whole range of different gaits intrinsically preprogrammed in locomotor CPGs. Many research approaches to modeling the mechanisms of multilegged locomotion are based on the interactions of coupled neural oscillators, which are described mathematically by dynamical system theory (Cohen et al. 1982; Bay and Hemami 1987; Scho¨ner 1990; Yuasa and Ito 1990; Wang and Rinzel 1992). These works thus need to use a specific continuous model of the target biological systems. As an alternative method, we present a discrete model approach by employing a special class of topology-independent graph dynamics to the modeling of the collective behavior of purely inhibitory neuronal networks (Yang and Franc¸a 1998; Franc¸a and Yang 2000). We will show that scheduling by edge reversal (SER) (Barbosa and Gafni 1989; Barbosa 1996) and its generalization, scheduling by multiple edge reversal (SMER) (Franc¸a 1994; Barbosa et al. 1996), are distributed algorithms that can be applied to predict or reproduce the interesting behavior of many biological oscillatory neuronal networks, especially CPGs. The methodologies on how to create oscillatory building block (OBB) networks exhibiting predefined oscillatory patterns described by target SMER dynamics and how such networks can be used to model biological motor systems much more simply and effectively are introduced in this paper. A 4n-node architecture representing 2n-legged animals, where n is the number of pairs of legs, proposed by Golubitsky et al. is adopted to avoid conjugate patterns (Golubitsky et al. 1998, 1999). However, unlike the group theoretic approaches, OBB networks are large-scaled, parallel, and distributed systems that can also be expressed as asymmetric Hopfield neural networks. It is also shown that OBB networks are capable of reproducing all gait patterns as well as confirming the efficiency of Golubitsky’s general model. For the sake of clarity, the five typical rhythmic gait

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patterns of the cockroach are chosen as case studies to illustrate this new approach. Nevertheless, it is also shown that this technique can be extended to reproduce most invertebrate and vertebrate rhythmic movements provided that they can be described topologically. All types of OBB modules can be synthesized into custom hardware (ASICs) with the help of field programmable gate array (FPGA) technologies (Yang and Franc¸a 1999). A general-purpose robot can thus be efficiently constructed with this control strategy if a suitable hardware platform is supplied. This paper is organized as follows. SER and SMER distributed algorithms are briefly depicted in Sect. 2. Section 3 outlines the characteristics of discrete and analog OBB modules with the instantiation of discrete OBB modules for diverse hexapodal gait rhythms. A general locomotion architecture and a case study of a quadrupedal gait are then presented based on an analog version of OBB modules in Sect. 4. Section 5 concludes the paper. 2 Inhibitory neuronal networks as neighborhood-constrained systems 2.1 Scheduling by edge reversal (SER) Consider a neighborhood-constrained system comprised of a set of processes and a set of atomic shared resources represented by a connected graph G ¼ ðN ; EÞ, where N is the set of processes and E the set of edges defining the interconnection topology. An edge exists between any two nodes if and only if the two corresponding processes share at least one atomic resource. SER works in the following way: starting from any acyclic orientation x on G there is at least one sink node, i.e., a node that has all its edges directed to itself. All sink nodes are allowed to operate while other nodes remain idle. This obviously ensures mutual exclusion at any access made to shared resources by sink nodes. After operation a sink node will reverse the orientation of its edges, becoming a source and thus releasing the access to resources to its neighbors. A new acyclic orientation is defined and the whole process is then repeated for the new set of sinks (Barbosa and Gafni 1989; Barbosa 1996). Let x0 ¼ gðxÞ denote this greedy operation; SER can be regarded as the endless repetition of the application of g(x) upon G. Assuming that G is finite, it is easy to see that eventually a set of acyclic orientations will be repeated defining a period of length p. This simple dynamics ensures that no deadlock or starvation will ever occur since at every acyclic orientation there is at least one sink, i.e., one node allowed to operate. Also, it is proved that inside any period every node operates exactly m times (Barbosa and Gafni 1989; Barbosa 1996). SER is a fully distributed graph dynamics algorithm. A very interesting property of this algorithm lies in its generality in the sense that any topology will have its own set of possible SER dynamics. Figure 1 illustrates the SER dynamics.

Fig. 1. A graph G under SER, with m ¼ 1, period length p ¼ 3

2.2 Scheduling by multiple edge reversal (SMER) SMER is a generalization of SER where prespecified access rates to atomic resources are imposed on processes in a distributed resource-sharing system that is represented by a multigraph MðN ; EÞ. Unlike SER, with SMER a number of oriented edges can exist between any two nodes. Between any two nodes i and j , i; j 2 N , there can exist eij unidirected edges, eij  0. The reversibility of node i is ri , i.e., the number of edges that shall be reversed by i toward each of its neighboring nodes, indiscriminately, at the end of the operation. Node i is an ri -sink if it has at least ri edges directed to itself from each of its neighbors. Each ri sink node i operates and reverses ri edges toward its neighbors, the new set of ri sinks will operate, and so on. Like sinks under SER, only ri sink nodes are allowed to operate under SMER. It is easy to see that with SMER, nodes are allowed to operate more than once consecutively. The following lemma states a basic topologic constraint toward the definition of M, where gcd is the greatest common divisor and fij is the sum of the greatest multiple of gcdðri ; rj Þ that doesn’t exceed the number of shared resources oriented from ni to nj , and from nj to ni , respectively, in the initial orientation. Lemma 1 (Barbosa et al. 1996; Franc¸a 1994) Let nodes i and j be two neighbors in M. If no deadlock arises for any initial orientation of the shared resources between i and j, then maxfri ; rj g eij ri þ rj  1 and fij ¼ ri þ rj  gcdðri ; rj Þ. ( It is important to know that there is always at least one SMER solution for any target system’s topology having arbitrary prespecified reversibilities at any of its nodes (Barbosa 1996). In the next sections, SMER will be employed on the construction of artificial CPGs by implementing oscillatory building blocks (OBBs) as asymmetric Hopfield-like networks, where operating sinks can be regarded as firing neurons in purely inhibitory neuronal networks. 3 Analog and discrete OBB modules and network The ensured mutual exclusion activity between any two neighboring nodes coupled under SMER suggests a scheduling scheme that resembles anticorrelated firing activity between inhibitory neurons exhibiting postinhibitory rebound (PIR), a neuronal mechanism underlying locomotion and other rhythmic activities (Wang and Rinzel 1992). The discrete and analog versions of

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Fig. 2. One possible scheme of firing circulation patterns of building blocks. a Four possible configurations for medium-speed gait pattern. b Six possible configurations for slow-speed gait pattern

SMER-based OBBs can be customized for different rhythmic patterns, where discrete OBB modules are built by directly adopting the SMER algorithm while an asymmetric Hopfield neural network is employed for implementing analog OBB modules. 3.1 Discrete OBB modules Instead of modeling electrophysiological activities of interconnected neurons based on membrane potential functions, we build an artificial CPG network with SMER-based OBBs for the collective behavior of a neuron set. The representative cases of three gait patterns of a cockroach, i.e., metachronal, mediumspeed, and fast-speed locomotions according to the prototypes of Collins and Stewart (Collins and Stewart 1993), are analyzed by using the discrete OBBs. As the locomotion proceeds, the gait pattern is composed of the collective behaviors of the states of six legs. In this paper, each leg is simplified as having one degree of freedom with a pair of flexor and extensor, though any degrees of freedom in a leg are feasible by assuming more complexity under the proposed strategy. From an anatomical point of view, cockroach leg movement is driven by the flexor and extensor motor neurons. The flexor lifts a leg from the ground, while the extensor does the opposite. This procedure can be well imitated with an OBB module by taking neurons i and j in the module as flexor and extensor, respectively (see Fig. 2). There is a timing relationship between the pair of flexor and extensor for the different locomotive speeds. As a cockroach escapes more swiftly, the firing duration of the cockroach’s extensor (corresponding to stance) will decrease drastically, while the firing duration of its flexor (corresponding to swing) remains basically unchanged, which is consistent with the biological experiments (Pearson 1976). This insight indicates that hexapodal speed is determined largely by the extensor firing, i.e., the time duration of a leg contacting the ground. The state transition of each leg and the corresponding phase relations among six legs (see Fig. 3) are important to simulate the gait model since the phase circulation can be represented by the circulation of OBB modules. In the case of a cockroach’s fast-speed gait pattern, the simple SER-based OBBs can be applied to initiate oscillation and coordinate the movement of the six legs

Fig. 3. Movement circulation sequences and phase relations among six legs of a cockroach, each leg represented by one electrically compact nodes. a Metachronal rhythm. b,c Slow-speed gait. d Medium-speed gait. e Fast-speed gait. L1 is the first left leg, R1 is the first right leg

(see Fig. 3e). For the more complicated rhythmic leg movements of metachronal and medium speed, in which the firing of neighboring nodes is not exactly out of phase and some phase overlaps exist, a two-step process should be followed to construct the SMERbased OBBs. Step 1. Determine the periodic movement of a gait pattern and construct the circulation of its corresponding OBBs. Step 2. Organize the artificial CPG network with different building block configurations, i.e., choose a suitable OBB from the circulation of OBB modules for each leg. In order to formulate the metachronal movement of a six-legged cockroach, an appropriate OBB configuration in the pattern circulation (see Fig. 2b) is chosen for each of six legs, respectively, according to the corresponding phase relationship presented in Fig. 3a. The coordinated patterns of the cockroach’s metachronal gait can thus be generated by different SMER-based OBB modules in their appropriate configurations. This process is generalizable to modeling the other gaits.

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The OBB network underlying a cockroach’s metachronal gait is a free-run artificial CPG consisting of an OBB module for each leg. The gait model operates asynchronously without a global clock. It is the OBB module rather than the OBB network that is governed by a SMER algorithm. The mutually inhibitory relation between the coupled flexor-extensor pair and the phase relation between the neighboring flexor-flexor and extensor-extensor pair are assumed so that the OBB modules are solely responsible for gait pattern generation and transition. The modulation of coupling conditions between the OBB modules and the internal parameters of an individual module can greatly alter network operation, even generate a totally new network (Getting 1989). In our model, this kind of modulation is assumed to be induced by consciousness signals from the central nervous system (CNS) through the conditioned/unconditioned reflex or the alteration of intrinsic characteristics of the OBB module itself. We will treat this kind of modulation and the subsequent gait transition in another paper. It is common sense that an animal’s locomotion behavior is a continuous procedure. The present artificial CPG based on discrete OBB modules generates some typical observations for these continuous, highdimensional waves (the number of wave dimensions is the number of animal legs) that are the snapshots with at least one leg supporting the body substantially and, ignoring all other snapshots, that may contribute little for the model retrieval. Since the high-pass-filter characteristic, which is a common property observed in every real neuron (Bassler 1986; Matsuoka 1987), may be naturally simulated by the hysteretic phenomena of mechanical oscillation, we believe that the pulse stream generated by the artificial CPG will lead to more smooth, continuous waveforms for behavioral simulation after the pulse stream is delivered to a mechanical platform. Four snapshots of a graphic experiment on cockroach’s medium-speed gait are shown in Fig. 3d; the rhythmic order is: ðL1R3ÞR2ðL3R1ÞL2 . . .. 3.2 Dynamic properties of analog OBB modules The SMER algorithm can be implemented schematically using a network like the Hopfield model (Hopfield and Tank 1985) with some modifications such as nonglobal connections based on the interconnecting topology of a locomotion system. Like the dynamics of cellular neural networks (Chua and Yang 1988a, b), the input and output voltages of each node in an OBB network are normalized to digital low or high level while the internal potential is continuous within the normalized interval [0, 1]. The SMER-based OBB modules can be classified into two complexity levels, namely, simple and composite. The simple OBB modules consist of only two interconnected nodes with prespecified reversibilities; the composite ones may contain an arbitrary number of cells interconnected with any topology. Both types of OBB modules follow Lemma 1 for initial shared resources arrangement and configuration to avoid possible abnor-

mal operation, e.g., deadlock or starvation, during oscillation. 3.2.1 Dynamics of simple OBB modules. Now consider a submultigraph of MðN ; EÞ, namely, M ij , having a pair of coupling nodes ni and nj with ri and rj as their reversibilities, respectively. In the SMER-based, simple OBB module, the postsynaptic membrane potential (PSP) i of each neuron i at k instant, Mpsp ðkÞ, depends on three i factors, i.e., its former PSP Mpsp ðk  1Þ, the impact of its j coupled counterpart neuron vout ðk  1Þ, and the negative feedback of neuron i itself viout ðk  1Þ, without considering the external impulse. The selection of system parameters, such as the neuron thresholds and synapse weights, are crucial for modeling. In our model, let r ¼ maxðri ; rj Þ and r0 ¼ hðrÞ, where h is a function of highest integer level and multiplying it by 10, e.g., if ri ¼ 21 and rj ¼ 151 then r0 ¼ hðrÞ ¼ hðmaxð21; 151ÞÞ ¼ hð151Þ ¼ 103 . Hence we can further design the thresholds of neurons i and j hi , hj and their synapse weights wij , wji as the following: 8 hi ¼ maxðri ; rj Þ=ðri þ rj  gcdðri ; rj ÞÞ > > < w ¼ maxðr ; r Þ=r0 ij i j ð1Þ h ¼ ðminðr ; rj Þ  1Þ=ðri þ rj  gcdðri ; rj ÞÞ > j i > : wji ¼ minðri ; rj Þ=r0 We arrange the system parameters by comparing two nodes’ reversibilities; if ri > rj , then we have hj > hi and wij > wji . This arrangement scheme ensures that the behavior of SMER-based OBB modules is consistent with its original SMER algorithm, i.e., a node with smaller reversibility (correponding to a neuron with lower threshold in an OBB module) will oscillate at a higher frequency than its companion does. The difference equation of this system can be formulated as follows: each neuron’s self-feedback strength is wii ¼ wij , wjj ¼ wji , respectively, and the activation function is the sigmoidal Heaviside type. It is worth noticing that k is the local clock pulse of each neuron, as a global clock is not necessary.  Mi ðk þ 1Þ ¼ Mi ðkÞ þ wji vj ðkÞ þ wii vi ðkÞ ð2Þ Mj ðk þ 1Þ ¼ Mj ðkÞ þ wij vi ðkÞ þ wjj vj ðkÞ where  vi ðkÞ ¼ maxð0; sgnðMi ðkÞ  hi ÞÞ vj ðkÞ ¼ maxð0; sgnðMj ðkÞ  hj ÞÞ

ð3Þ

We consider the designed circuit as a conservative dynamical system in the ideal case, i.e., the total energy is constant, no loss or complement is allowed, the sum of two cells’ PSP at any given time is normalized to 1. It is not difficult to see that this system has the capability of self-organized oscillation with the firing rate of each neuron arbitrarily adjustable. However, like most dynamic systems, our model also has a limit in the dynamic range, i.e., there exists a singular point as each cell’s PSP equals its threshold; in this way, the system may involve in another different oscillation behavior or even halt.

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Within its dynamic range, the properties of the OBB module show that this design is suitable for implementation of the SMER algorithm. Proposition 1 The circuit representing the fundamental two-node SMER system is a starvation- and deadlock-free oscillation system. Proof: Let us assume the circuit is a conservative system without damping and noise, and we also ignore the delay of the synapse. Since the energy of this two-node, ideal system is constant, we suppose that the normalized system potential is 1 and the normalized potential of neuron i is a 2 ½0; 1 , so the potential of neuron j should be b ¼ 1  a 2 ½0; 1 . From Eq. 5 we get Mi ðk þ 1Þþ Mj ðk þ 1Þ ¼ Mi ðkÞ þ Mj ðkÞ; this also means that the system energy is constant. Hence, if a > hi , then b ¼ 1  a < 1  hi ¼ hj . Similarly, if a < hi , we can get b > hj . It is therefore impossible for two coupled neurons to fire simultaneously. It is intuitive to understand the circuit mechanism such that if any neuron fires, its PSP will keep decreasing by the value of coupling strength with regard to its neighbor per local pulse until its PSP passes down through its threshold and the cell becomes idle. Meanwhile, the PSP of the idle cell keeps increasing by the same value per local pulse until it begins firing. This process will keep repeating, and it guarantees the starvation-free and deadlock-free oscillating mechanism of this circuit for a two-node SMER system; the PSP of each cell is continuously changeable within ½1; 2 . ( The simple OBB module has another prominent property that is essential for designing a CPG system, i.e., the periodicity of the oscillation system. Since the convergence procedure depends strongly on the neurons’ reversibilities and initial conditions, which are random and impossible to determined a priori, a general proof of the following proposition is an analytical derivation by

calculation on the two coupled neurons’ PSP, threshold, and weight values, as given in the appendix. Proposition 2 The simple SMER-based OBB module is a stable and periodic oscillation system no matter what initial potential its neuron may have. A computer simulation of this simple OBB module is conducted on the predefined neuron reversibilities ri ¼ 4 and rj ¼ 3 with the arbitrary initial PSP as Mi ð0Þ ¼ 0:2 and Mj ð0Þ ¼ 0:8 (see Fig. 4). The numerical integration is made by the fourth-order Runge-Kutta method, and the high-frequency bioelectronic activities in the highpotential level are simulated by adopting membrane conductances and capacitors for each neuron. 3.2.2 Dynamics of a composite OBB module. The composite OBB module is a generalized version of the aforementioned simple OBB module in the sense that the composite OBB module is a system consisting of a number of simple OBB modules. Suppose a multigraph MðN ; EÞ containing a set of m nodes N ¼ n1 ; n2 ; :::nm and a set of edges E ¼ < 0j1 >ij , where 8i; j 2 ½1; m , which define the connection topology of this multigraph by using < 1 >ij and < 0 >ij for with and without an edge between nodes i and j, respectively. Each node i has its own reversibility ri . There are eij shared resources on the corresponding edge < 1 >ij , with their number and configuration obeying Lemma 1. Unlike the simple OBB module in which a node is coupled with another unique node to form a two-node model, the main structural feature of a composite OBB module is that it has at least one node coupled with at least two additional nodes. By dissecting a composite OBB module into various simple ones, each node of the composite OBB module is split into the corresponding copies (the set of copies shares the same local clock of

Fig. 4. The computer simulation of a simple OBB module based on a SMER case; the predefined node reversibilities are ri ¼ 4 and rj ¼ 3 with the arbitrary initial PSP as Mi ð0Þ ¼ 0:2 and Mj ð0Þ ¼ 0:8. a The state circulation of a simple OBB module, where steps k ¼ 0 to 6 form a period, the output states and internal membrane potentials are signified by node i ¼ vi ðkÞ=Mi ðkÞ, node j ¼ vj ðkÞ=Mj ðkÞ corresponding to two coupled, discrete nodes in the SMER case. b1 Oscillating waveform of node i and b2 oscillating waveform of node j

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their maternal node). Each copy is a component of a simple module, and the number of copies is twice as many as the number of edges in the composite OBB module. We then terminologically treat a node in a composite OBB module as a macroneuron and a node’s copy as its clone. Their definitions are also given as follows. Definition 1 A macroneuron is defined to be a node i that satisfies 8 i 2 N , where N is the set of nodes in multigraph MðN ; EÞ. (

Definition 2 A clone, which has independent reversibility and represents the coupling characteristics of its host macroneuron with one of the neighboring macroneurons, is a unique component of its maternal macroneuron. ( According to the principle of SMER, a macroneuron of the composite OBB module operates in a ‘‘whole-ornone’’ mechanism, i.e., it will fire if and only if all its clones fire. The outputs of a macroneuron’s clones are multiplied logically to denote the state of the macroneuron. The clones keep checking the state of their maternal macroneuron. If the macroneuron is firing, then its clones operate; otherwise they remain idle. Fundamentally, a composite OBB module operates on the basis of its component simple OBB modules. A schematic diagram of a composite OBB module is shown in Fig. 5, which is the modification of an asymmetric Hopfield net in the sense that additional feedback from the logical sum combination of two outputs of the coupled macroneurons is provided to control the transition of the shared resources on each corresponding edge. As the generalized version, a composite OBB module can represent a more complicated oscillating neuronal network and reproduce more rhythmic patterns than a simple one. Since it is impossible to assign a unified threshold to a macroneuron that may have more than one independent clone, the usual way to analyze this kind of modules is, as we said in the introduction, to dissect them into the subsystems of simple modules where Eqs. 1–3 are also applicable. The output of a macroneuron, ni , is then determined by all its clones.

Fig. 5. Two equivalent ways to illustrate the SMER algorithm. ri ¼ rm ¼ rn ¼ 1; rj ¼ 2; rk ¼ 3. (Left) Original SMER description. (Right) An alternative description for the composite OBB module with the macroneurons and their clones

Vi ðkÞ ¼

n Y

vji ðkÞ

ð4Þ

j¼1

where 8 i 2 N , i 6¼ j and vji are the output of clone j of the macroneuron i, which couples with a corresponding clone of another macroneuron. The superscript sequence 1; 2; . . . ; n is the clone number of a macroneuron ni with n m  1. The key issue for achieving the desired rhythmic patterns from a composite OBB module is to choose the initial PSP values properly for its simple OBB modules, e.g., no clone should be idle if its macroneuron is firing initially. Within an appropriate scope, the selection of initial PSP values is arbitrary. After an initial duration whose length is determined by the different selections of initial PSP values, the system will oscillate periodically.

4 A general locomotion CPG architecture It is widely believed that animal locomotion is generated and controlled in part by CPGs, which are networks of neurons in the CNS capable of producing rhythmic output. Current neurophysiological techniques are unable to isolate such circuits from the intricate neural connections of complex animals, but the indirect experimental evidence for their existence is strong (Grillner 1975; Pearson 1976; Stein 1978; Grillner 1985; Golubitsky et al. 1998, 1999).

4.1 The architecture The general architecture of a locomotion system may consist of three parts, i.e., the oscillation system as driving motor, the CPGs as pattern source, and the CNS as command source. The first two parts formulate the proposed locomotion architecture in this paper and can thus be constructed by a model with reciprocally inhibitory relationship between the so-called flexor and extensor motor neurons in each leg as driving motor and the general locomotor CPGs as pattern source. The mechanism of the CNS is beyond the scope of this paper. As the core of the proposed locomotion architecture, the CPG converts different spiritual signals from the CNS to the bioelectronic-based rhythmic signals for driving the flexor and extensor, respectively. These motor neurons are essentially relaxation oscillators, i.e., a pair of flexor and extensor forms a bistable system in which each neuron switches from one state to another when one of the system’s two internal thresholds is reached and back when another threshold is reached again. For a 2n-legged animal the proposed model of general locomotor CPGs has the topology of a complete graph 2 (see Fig. 6). Each MðN ; EÞ with kN k ¼ 4n and kEk ¼ C4n of the upper-layer macroneurons corresponds to a flexor, while the lower-layer ones correspond to an extensor. Different gaits may have different couplings within this unified architecture in the sense that some couplings are

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Fig. 6. A general locomotor CPG architecture of 2n-legged animal; each macroneuron couples with all of the remaining 2n  1 macroneurons, which is not shown explicitly, kNk ¼ 4n and 2 kEk ¼ C4n ¼ 4n!=2!ð4n  2Þ!; the upper is the flexor layer while the lower is the extensor layer

vanced animals may have fewer neural processing activities besides the CNS-CPGs signals. The of a macroneuron Vi has the formula Q output j j Vi ¼ 4n1 j¼1 vi , where 8 i 2 N , i 6¼ j and vi are the output of clone j of macroneuron i, which couples with the corresponding clone of another macroneuron. It is known that locomotion speed is determined mainly by both coordinated phase relation and duty factor, which is the relative proportion of flexor firing in one period. As a legged animal’s gait speed increases, the extensor firing time will decrease drastically while the firing time of the flexor neuron is basically constant (Pearson 1976). The duty factor can be modified by changing the reversibilities of two coupled nodes.

retained while the rest of the couplings are blocked, or the blocked couplings are recovered during gait transition. It is reasonable for the gait transition to be controlled by the biological signals from the animal’s CNS for avoiding risk or from its own willingness; the amount of signal flow that is equal to the existing total coupling number depends on the number of animal legs. For those specific models derived from the unified model, the bipedal, quadrupedal, hexapodal animals may have, respectively, 4, 16, and 30 bits of parallel signals from their CNS to CPGs. One cannot determine the advancement level of an animal according to the number of descending CNS signals since the meaning of the signal number is twofold. First, even a centipede, which is a relatively low-level insect, may have many CNS control signals, yet still they occupy only a very small percentage of neural processing; second, more advanced animals such as bipeds have many neural processing activities besides the CNS-CPG controlling signals, while less ad-

A computer-simulated example of the quadrupedal trot gait was conducted employing the resulting circuit and respecting the predefined cells reversibilities. The macroneurons in the upper flexor layer are the same as those shown in Fig. 6, while the macroneurons in the lower extensor layer are marked with 5, 6, 7, 8 instead of 2n þ 1, 2n þ 2, 2n þ 3, 2n þ 4, respectively, with the rest of the macroneurons eliminated. The simulation result of a quadrupedal trot gait is shown in Fig. 7. The criteria for choosing the initial PSP are important and consistent with those of the original definition of the SMER dynamics. There may be a convergence delay at the beginning of operation until a periodic behavior is achieved since a random choice of the initial PSP within

Fig. 7. Simulation of quadrupedal trot gait with asymmetric Hopfield-like net. Macroneurons 1, 2, 3, and 4 correspond to flexor muscles of right hind, left hind, right front, and left front, respectively; they are located in the upper layer of the CPG architecture shown in Fig. 6. Macroneurons 5, 6, 7, and 8 are the auxiliary neurons located in the bottom layer and conjugate to the upper layer. The predefined

reversability of a macroneuron in each coupling direction is rij ¼ 1, where 1 i; j 8 and i 6¼ j, under the mechanism of a simple OBB module governed by a SER algorithm. The overall activation state of a macroneuron depends on the multiplication of its activation states in all coupling directions. The oscillating waveforms of macroneurons 1  8 are shown from Eq. 1 to 8, respectively

4.2 Simulation of a quadrupedal gait

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the prescribed scope is carried out. The output of each macroneuron is the product of all its clones’ outputs connecting with this macroneuron’s neighbors, i.e., a macroneuron will fire if and only if its clones connecting all its neighbors are firing simultaneously. An additional peripheral feedback control neural circuitry is given based on the SMER algorithm, which requires that all the shared resources on an edge not reverse (i.e., keep its original position of last clock pulse) unless the output of one out of two coupled macroneurons at the terminals of this edge is firing. Obviously there is an initial selforganizing procedure as 3 ! 4 ! 3 ! ð14Þ7 ! ð23Þ8 ! ð14Þ7 ! ð23Þð58Þ Ð ð14Þð67Þ. The matrices of the thresholds (hij ), coupling weights (wij ), and initial membrane potentials (Mð0Þij ) of all neurons in the quadrupedal gait achitecture for the trot gait is as follows. The order of column and row in the matrices is from macroneuron 1 to 8. 1 0 0 0 1 0 1 0 0 0 B1 0 0 1 0 0 0 0C C B B0 0 0 0 0 0 1 0C C B B0 0 1 0 0 0 0 0C C B hij ¼ B C B0 0 0 1 0 0 0 0C B0 1 1 0 1 0 0 0C C B @0 1 0 0 1 0 0 0A 1 0 0 1 0 1 1 0 0

0:1 B 0:1 B B 0:1 B B 0 wij ¼ B B 0:1 B B 0 B @ 0 0:1 Mð0Þij ¼ 0 0 B 1:07 B B B 0:16 B B 0 B B B 0:18 B B 0 B B @ 0 1:02

0:1 0:1 0 0:1 0 0:1 0:1 0

0:07 0 0 0:16 0 1:16 0:96 0

0:1 0 0:1 0:1 0 0:1 0:1 0

1:16 0 0 1:03 0 1:08 0:18 0

0 0:1 0:1 0:1 0:1 0 0 0:1

0 1:16 0:03 0 1:14 0 0 1:16

0:1 0 0 0:1 0:1 0:1 0:1 0

0:82 0 0 0:14 0 1:04 0:91 0

0 0:1 0:1 0 0:1 0:1 0 0:1

0 0:16 0:08 0 0:04 0 0 0:92

0 0:1 0:1 0 0:1 0 0:1 0:1

0 0:04 0:82 0 0:09 0 0 1:09

1 0:1 0 C C 0 C C 0:1 C C 0 C C 0:1 C C 0:1 A 0:1

1 0:02 0 C C C 0 C C 0:16 C C C 0 C C 0:08 C C C 0:09 A 0

network (analog) version. It is not difficult to understand the proposed model considering that the locomotion of a legged animal is actually a period during which parts of its body weight are repeatedly assigned to its legs, just like the resources being shared between nodes in the aforementioned distributed algorithms. Theoretically, with this general methodology one can reconstruct almost all the gait patterns of legged animals, as introduced by Collins and Stewart (1992, 1993) and Collins and Richmond (1994) using the symmetrybreaking Hopf bifurcation theory. In order to build a real locomotion system, it would be convenient to adopt a customized hardware platform with reconfigurable features to integrate various types of OBB modules in the construction of a complete artificial CPG model. We have shown that the concurrent and signal-driven natures of this distributed, macroscopic approach match well the characteristics of the high-level design methodology of customised hardware (Yang and Franc¸a 1999). This locomotion model is thus tailor-made for discrete and digital implementation and may provide a novel direction on combining the theories of artificial neural networks with robotics design methods. Acknowledgements. This work was partially supported by CNPq, the Brazilian Research Agency, under support number 143032/968. We are grateful for the helpful discussions with Prof. V.C. Barbosa, Dr. A.E. Xavier, Dr. M.S. Dutra, and Dr. A.F.R. Ara ujo. The donations of FPGA hardware and software from XILINX Incorporation under the order No. XUP2930 and XUP3576 are also highly appreciated.

Appendix For a two-node, discrete SMER case, its periodical oscillation is apparent and has been proven (Barbosa et al. 2001; Franc¸a 1994). However, an asymmetric Hopfield neural net with an embedded SMER mechanism is a continuous behavior system in its internal state. Moreover, the initial PSP of each node is randomly chosen within a normalized scope ½1; 2 . Should this net exhibit a periodical oscillation like its SMER prototype? Proposition 2 guarantees the net’s behavior. In order to prove proposition 2, we need a prerequisite Lemma 2. Lemma 2 The limit of a sequence aN is zero if aN is a monotonically decreasing sequence and satisfying aN þ1 ¼ aN 1  baN 1 =aN c  aN , with the initial condition of a0 > a1 > 0.

ð5Þ Proof: Since aN is a monotonically decreasing sequence, i.e., 0 aN þ1 < aN , we have 5 Conclusion We have presented a novel approach to the modeling of the collective behaviors of biological locomotion. It was demonstrated that there is an equivalence between any discrete SMER-based OBB and its asymmetric Hopfield

lim aN ¼ b  0

n!1

If b > 0, there always exists a sufficiently big integer N such that aN  aN þ1 < b. However, from aN þ2 ¼ aN  baN =aN þ1 c  aN þ1

42

we have aN ¼ baN =aN þ1 c  aN þ1 þ aN þ2  aN þ1 þ b so the result is aN  aN þ1  b, which is contrary to aN  aN þ1 < b. Hence we can only have b ¼ 0. ( Proposition 2 The simple SMER-based OBB module is a stable and periodic oscillation system no matter what initial potential its neuron may have. Proof: Without loss of generality we suppose in the coupled two-node system, ri > rj , and initially node j is firing, i.e., Mj ð0Þ > hj , Mi ð0Þ < hi . No matter what initial PSP node j may have, after a convergence procedure, nodes i and j will arrive at a pulse when they are at their final firing stage (now suppose this stage is the initial stage): Mj ð0Þ 2 ðhj ; hj þ wji Þ, Mi ð0Þ 2 ðhi  wji ; hi Þ. After that, node i will fire and, according to Eq. 1, wij > wji , wij =wji ¼ a, node i fires for only one pulse and becomes idle again. From now on node j will fire from an interval from which it will always begin its firing. A four-step interval cycle is thus shown below. 1. Mj ð0Þ 2 ðhj ; hj þ wji Þ, Mi ð0Þ 2 ðhi  wji ; hi Þ 2. Mj ð1Þ 2 ðhj  wji ; hj Þ, Mi ð1Þ 2 ðhi ; hi þ wji Þ 3. Mj ð2Þ 2 ðhj þ ða  1Þ  wji ; hj þ a  wji Þ, Mi ð2Þ 2 ðhi  a  wji ; hi  ða  1Þ  wji Þ 4. Some convergence procedure is followed, and the system state returns to state I Apparently it is the case of the prerequisite lemma, so proposition 2 is proved. (

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