A SIMPLE MODEL OF A CENTRAL PATTERN GENERATOR FOR QUADRUPED GAITS

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1 A SIMPLE MODEL OF A CENTRAL PATTERN GENERATOR FOR QUADRUPED GAITS JEFFREY MARSH Humankind s long association with four-legged animals, wild as well as domesticated, has produced a rich vocabulary of words describing their size, shape, age, color, gender, and other physical attributes. In addition, there is a whole class for words referring to animal gaits. A gait is a repeating temporal pattern of leg movements used for locomotion. Common gaits of horses include the trot, canter, and gallop. In many animals, it is known that rhythmic motions are controlled not by signals directly from the brain, but by smaller collections of neurons in the spinal cord [5]. These neural networks are known as central pattern generators. Each neuron in such a network is referred to as a cell. In this project we consider a network of eight coupled cells which provides a model of a central pattern generator for the quadruped gaits walk, trot, pace, and canter [1, ]. 1. The Network The neural network we use is diagramed in Figure 1 where each cell is represented by a circle and the coupling from cell i to cell j is represented by an arrow from cell i to cell j. The different gaits are produced by changing the values of four coupling constants (α, β, γ, and δ) between the cells. Date: 6 May 1. Submitted in partial satisfaction of the requirements of the course Mathematics 548: Mathematical Modeling. Front 4 Right Left 5 Rear Figure 1. The eight-cell network used to model quadruped gaits. 1

2 JEFFREY MARSH The outputs of four of the cells in the network are each associated with the motion of one of the four legs of the quadruped. Cell 1 corresponds to the left hind leg (LH), cell to the right hind leg (RH), cell 3 to the left fore leg (LF ), and cell 4 to the right fore leg (RF ). The outputs of cells 5 through 8 are not considered here. We define an event called a footfall when the output of a given cell has positive slope and exceeds a certain threshold. For a given gait, this network models only the repeating pattern of footfalls in time. No attempt is made to model other leg motions such as joint angles or height of the foot above the ground. It has been shown using symmetry arguments [] that the smallest network that can model the gaits of walk, trot, and pace consists of eight cells, and it is hypothesized that two cells per leg are required because the flexor and extensor systems of muscles must be controlled separately.. The Cells Each cell is identical and is modeled as a system of two ordinary differential equations, the Morris-Lecar equations [4]. This system is widely used to describe the electrical behavior of neurons. The simplified Morris-Lecar equations are: v = g Ca m(v)(v 1) g L (v v L ) g K w(v v K ) + I ẇ = φτ(v)(n(v) w) where m(v) = 1 ( ( )) v v1 1 + tanh v n(v) = 1 ( ( )) v v3 1 + tanh v 4 and ( ) v v3 τ(v) = cosh v 4 Here v represents the electric potential (voltage) across the membrane of the neuron, and w represents the fraction of channels that are capable of transporting ions across the membrane. The other quantities appearing in the equations, φ, v 1, v, v 3, v 4, g Ca, g L, g K, v L, v K, and I, are treated as parameters. In this project we will be primarily concerned with v, as it represents the output of the cell. This system displays some very interesting behavior depending upon the choice of parameters. One to three equilibrium points in (v, w) phase space are possible. We choose a set of parameters (φ =., v 1 =., v =.4, v 3 =.3, v 4 =., g Ca = 3, g L =.6, g K = 1.8, v L = 1.8, v K =.8, and I = 1) for which the system has rather tame (i.e., boring) behavior. The vector field for this system with these parameters is shown in Figure. The two nullclines ( v = and ẇ = ) are shown. They intersect at (v eq, w eq ) =

3 CPG FOR QUADRUPED GAITS W V Figure. Vector field for the Morris-Lecar system. (.396,.541), which represents the system s single equilibrium point (a spiral sink). A cell whose output is momentarily grounded (i.e., the voltage is forced to zero) and then released follows the trajectory shown. The voltage versus time evolution of this trajectory is plotted in Figure 3. A single Morris-Lecar cell with these parameters does not oscillate, but decays to the equilibrium state. However, things get more interesting when we couple two or more cells. 3. The Coupling To explain how systems of N Morris-Lecar cells are coupled, we first define the state vector for the i th cell as x i = (v i, w i ) where i = 1,..., N. The dynamical system for an uncoupled single cell is then ẋ i = f(x i ). For a network of N coupled cells we have ẋ i = f(x i ) + j i α ij h(x j, x i ) Where α ij is the strength of the coupling from cell j to cell i. The coupling is called bidirectional if a ij = a ji. Also, h(x j, x i ) is a function that specifies the

4 4 JEFFREY MARSH V t Figure 3. Voltage evolution of a Morris-Lecar cell that has been momentarily grounded. form of the coupling. For linear diffusive coupling we have h(x j, x i ) = x j x i and for linear synaptic coupling we have h(x j, x i ) = x j. For example, a network of two cells with bidirectional linear synaptic coupling would be described by: ẋ 1 = f(x 1 ) + αx ẋ = f(x ) + αx 1 In this project we consider only linear synaptic coupling and assign four coupling constants as follows: The constant α determines the strength of the coupling only between v i and v j and only between cells in the front and cells in the rear (see Figure 1). Specifically, α couples v 1 to v 3, v 3 to v 5, v 5 to v 7, and v 7 to v 1 on the left side, and v to v 4, v 4 to v 6, v 6 to v 8, and v 8 to v on the right side. Constant β has identical cell assignments but couples w i to w j. Constant γ couples v i to v j but only between cells on the right and cells on the left. Specifically, γ couples v 3 to v 4, v 4 to v 3, v 7 to v 8, and v 8 to v 7 on the front, and v 1 to v, v to v 1, v 5 to v 6, and v 6 to v 5 on the rear. Constant δ has identical cell assignments but couples w i to w j. 4. The Gaits 4.1. Walk. In a walk, footfalls occur in the sequence LH, LF, RH, RF with approximately a quarter-period phase difference between successive footfalls. This gait is called symmetrical, that is, the footfalls of a pair of feet (fore or hind) are evenly spaced in time [3]. The walk is known as a four-beat gait because the cadence of footfalls has four distinct beats per period. The walk is used by most mammals for locomotion at slow speeds.

5 CPG FOR QUADRUPED GAITS 5 LH RH LF RF Figure 4. Voltages versus time for the walk gait. For this gait we assign the coupling constants as follows: α =.1, β =.1, and γ = δ = 1.. Non-zero initial conditions are v 1 () = 1.4. As mentioned above, the coupling for all networks considered here is linear synaptic. The eight-cell neural network gives us a system of 16 first-order non-linear differential equations, for which a numerical solution is easily found using Matlab. In Figure 4 we plot the voltage output versus time of the first four cells: v 1 (t) = LH, v (t) = RH, v 3 (t) = LF, v 4 (t) = RF. As can be seen, if we set a threshold around 1.9 or so, the spikes of the voltage curves neatly correspond to footfalls that are in the correct sequence and have approximately a quarter-period phase difference. 4.. Trot. For the symmetrical gait known as trot, diagonal pairs of legs strike the ground more-or-less simultaneously and there is approximately a half-period phase difference between the diagonal pairs. The sequence of footfalls can be represented as LH + RF, followed by LF + RH. The trot is a two-beat gait, as there are two distinct beats per period. This gait is used by most mammals for medium speed; some, such as those with short to medium legs, adopt no other gait. Coupling constants for the trot are: α = β = γ = δ =.6.

6 6 JEFFREY MARSH LH RH LF RF Figure 5. Voltages versus time for the trot gait. Solving the system of differential equations yields the voltage curves shown in Figure 5. Again we see for a threshold around 1.9, the spikes of the voltage curves correspond to footfalls that are in the correct sequence and have the correct phase relationship Pace. In a pace, legs on the same side (right or left) strike the ground more-or-less simultaneously with approximately a half-period phase difference between the right and left pairs. The pace is also a symmetrical gait and we may represent the sequence of footfalls by LF + LH, followed by RF + RH. The pace, like the trot, is also a two-beat gait. Camels, some horses, and large slender dogs use this gait. The coupling constants for the pace are: α = β =., and γ = δ =.. Non-zero initial conditions are v 1 () =.4 and w 1 () =.9. We numerically solve the system and obtain the voltage curves shown in Figure 6. These curves are not as nice and spiky as the pervious examples, but with a threshold of.8 we have the correct sequence and phases of footfalls Canter. A canter is not a symmetrical gait. In a canter, one diagonal pair of legs strike the ground more-or-less simultaneously, but the legs of the other diagonal pair are approximately a half-period out of phase. The

7 CPG FOR QUADRUPED GAITS 7 LH RH LF RF Figure 6. Voltages versus time for the pace gait. sequence of footfalls can be represented as LF, RH, LH + RF. The canter is a three-beat gait, and is employed by horses for medium to fast speeds. For the canter the coupling constants are: α =.17, β =., γ =.9, and δ = 1.. Unfortunately, for this gait we have to tweak the cell parameters, setting φ =. and g Ca = 8.. Non-zero initial conditions are: v 1 () =.4 and w 1 () =.3. Again we numerically solve the system and obtain the voltage curves shown in Figure 7. A threshold of 1.7 works nicely to generate the correct relative phases and the sequence of footfalls. Note that LH and RF and not exactly in phase, but they are very close. The cadence of the canter gait is immediately recognizable even to citydwellers who have never been in the presence of a real horse. We wanted to use the numerical solution to generate a sound file of this gait, but did not have the time to do so. 5. Final Remarks The systems of differential equations in this project were solved using Matlab. The Morris-Lecar equations are what is called a stiff system and we have found that for certain parameters and initial conditions, the Matlab ODE solver ode45 takes a very long time to cough up a solution.

8 8 JEFFREY MARSH LH RH LF RF Figure 7. Voltages versus time for the canter gait. Therefore, we have used the solver ode15s, which is specifically designed for stiff systems. Figure was produced using the Matlab program pplane7.m which is copyright by John C. Polking of Rice University. Matlab is copyright , The Mathworks, Inc. References 1. Pietro-Luciano Buono, Models of central pattern generators for quadruped locomotion: Secondary gaits, Journal of Mathematical Biology 4 (1), Pietro-Luciano Buono and Martin Golubitsky, Models of central pattern generators for quadruped locomotion: Primary gaits, Journal of Mathematical Biology 4 (1), Milton Hildebrand, The quadrupedal gaits of vertebrates, BioScience 39 (1989), Catherine Morris and Harold Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical Journal 35 (1981), J. D. Murray, Mathematical biology, first ed., Biomathematics, no. 19, p. 58, Springer Verlag, Berlin, 1989.