How can neural systems
change their behaviour?
(More on stable states...)
State parameters of a coin (the dimensions in which the state of the coin can vary)
| Rotation around the x axis |
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| Rotation around the y axis |
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| Rotation around the z axis |
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State space of a coin
- this represents all the possible
orientations of the coin in space.
Stable states of a coin
Not all of the possible states (orientations) of the coin are equally likely. Some states are far more likely than others, such as when the coin is lying flat with heads or tails facing up. These are stable states because the coin tends to return to these positions after small perturbations. Those states which lead to a stable state (such as balancing on one edge at a 45 degree angle) are said to fall within the basin of attraction of the stable state. States with large basins of attraction, or large attractors, occur more frequently than those with small basins of attraction because it is far more likely that the coin will pass through one of the transitional states of a large attractor than a small one. In some cases, such as when the coin is spinning on its edge, the state is dynamically stable for as long as the motion continues. If the coin slows down to the point that it falls into the basin of attraction for heads or tails it will come to rest in a new stable state.
| heads (large attractor) |  |
| tails (large attractor) | |
| edge (very small attractor) |  |
| spinning (dynamically stable) |  |
The quadrupedal gait model
Treadmill studies in cats show that stepping patterns can be generated at the level of the spinal cord. These motor patterns can be activated or modulated by input from the mesencephalic locomotor region in the brain.
The model network - Canavier
et al. (1997)
The network consists of four (bi-stable) oscillators coupled to
form a ring circuit. The behaviour of each oscillator can be modulated
by an input parameter (ISTIM), which represents the input from the mesencephalic
locomotor region.
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| Each element of the ring circuit is a oscillator, which makes an inhibitory synapse onto the next element in the circuit. This is not the final circuit used in this study. | In order to make the correct association between the left and right hind and front limbs of a quadruped, two elements of the circuit are flipped. |
Once the model was established the ISTIM parameter was varied to allow the model to explore the possible areas of state space available. The output of the four oscillators were compared to quadrupedal gaits where each oscillator represented a particular limb. (See figure below.)
Under these conditions the network exhibited several stable states with the correct phase relationships for quadrupedal gaits (gallop, walk, bound).
In addition to simply varying the ISTIM parameter, mimicking varying
input from the brain, the authors also performed a Phase Response
Curve (PRC) analysis. To do this they calculated the combined
response of the circuit when the action of each oscillator perturbs
the next oscillator in the cycle. In the figure below, P0 is the normal
period of the oscillator, and P1 is the new period when the oscillator is perturbed
by a transient inhibition from the previous oscillator in the
ring circuit.
Using this kind of analysis the authors discovered numerous recognisable
quadrupedal gaits, all of which represent a kind of dynamic stability
because the oscillators are constantly perturbing each other as
the the pattern repeats itself. The table below shows the gaits
revealed by PRC analysis with the relative phase angle for each
oscillator representing the phase angle between different legs.
When the authors introduced 'bursting' states for one or more of the bi-stable oscillators they found some really cool, unusual gaits.
Epilogue
The ring circuit model network is an interesting way to simulate quadrupedal gaits, but the real mechanism is still unknown. Some points in favour of the ring circuit model: