A model of antennal wall-following and escape in the cockroach

Abstract

Cockroaches exploit tactile cues from their antennae to avoid predators. During escape running the same sensors are used to follow walls. We hypothesise that selection of these mutually exclusive behaviours can be explained without representation of the stimulus or an explicit switching mechanism. A neural model is presented that embodies this hypothesis. The model incorporates behavioural and neurophysiological data and is embedded in a mobile robot in order to test the response to stimuli in the real world. The system is shown to account for data on escape direction and high-speed wall-following in the cockroach, including the counter-intuitive observation that faster running cockroaches maintain a closer distance to the wall. The wall-following behaviour is extended to include discrimination of tactile escape cues according to behavioural context. We conclude by highlighting questions arising from the robot experiments that suggest interesting hypotheses to test in the cockroach.

Appendix

The neural simulation, written in C and assembler, ran directly on the Khepera robot’s 16 MHz MC68332 processor. The main loop updated all neurone and synapse units at 500 cycles/s, providing minimum time resolution of 2 ms. Pseudo-code for the neural model is given in Algorithm 1.

Algorithm 1 Neural model pseudo-code

Neurone model

Neurones were modelled as a single compartment with a single variable, the membrane potential (MP). Difference equations were defined that governed the change in MP for all possible values. In general, these acted to return the MP to its resting level (0), either by non-linear hyperpolarisation recovery, linear leakage, or reseting to a recovery value (following a spike). Attached to the neurone compartment were a dendrite and axon (see Fig. 13). The dendrite was modelled as an accumulation buffer that summed synaptic input over each time step. The axon represented the recent neural firing pattern as a pulse train. At each update cycle the spike pulses were shifted by one step along the axon, where they could be detected by the synapse components. The update function for a single neurone unit at each time step is given as pseudo-code in Algorithm 2. Full source code and parameterisation used by the model can be found in Chapman (2001).

Fig. 13

Diagrammatic representation of a neurone component showing dendrite, compartment, axon and synapses. At each simulated time step, incoming neurotransmitter is accumulated in the dendrite. The dendritic activation is then added to the membrane potential of the leaky compartment. If the membrane potential reaches threshold, the neurone fires a single spike. This is represented by setting a bit at the spike initiation zone (SIZ) in the axon. At each time step the axonal bit pattern is shifted one place to the right. Each synapse monitors a specific site on the axon for outgoing spikes

Algorithm 2 Neurone component pseudo-code

Parameters

Decay rate (16 bit signed integer, timeconst) When MP was sub-threshold, but above resting potential (zero), this amount was subtracted from the membrane potential at each time step. The value can be positive (leaky integration), negative (spontaneous firing) or zero (non-leaky integration).

Hyper-polarisation recovery rate (8 bit unsigned integer, hyperconst) When MP was negative its magnitude was halved this number of times at each time step, using an arithmetic right bit shift. In all the reported work this parameter was held constant at one.

Threshold level (16 bit signed integer, threshold) When MP was greater than this value the neurone generated an action potential (spike) and MP was reset to the recovery level.

Recovery level (16 bit signed integer, recovery) When the neurone fired the MP was reset to this value.

Variables

Synapse potential (16 bit signed integer, synpot) An accumulator that summed all neurotransmitter input to the neurone during each time step. The synapse potential was reset to zero at the end of each time step.

Membrane potential (16 bit signed integer, activation) The membrane potential (activation level) of the neurone at the current time step. The MP was adjusted according to

$$ {\text{MP}} (t + 1) = \left\{ \begin{array}{ll} \text{MP}(t)/2 + {\text{synpot}}(t) & \text{for}\ {\text{MP}}(t) < 0 \\ {\text{MP}}(t) - {\text{timeconst}} + {\text{synpot}}(t) & {\text{for}}\ 0 \le {\text{MP}}(t) \le {\text{threshold}} \\ {\text{recovery}} & {\text{for}}\ {\text{MP}}(t) > {\text{threshold}} \end{array} \right\} $$

(1)

On any cycle during which the neurone fired (i.e. MP(t) > threshold) the bit at the Spike Initiation Zone (SIZ) of the axon was set.

Axon pulse train (16 bit unsigned integer, firing) Contained a binary string representing the firing pattern of the neurone over the previous 16 cycles. New spikes were recorded by setting the bit at the SIZ, usually bit 15. At each time step the pattern was logically bit shifted one step to the right, thus removing the oldest bit and freeing up the SIZ for the current timestep.

Synapse model

Static synapse components acted as transmitters of excitation or inhibition from a specified axonal bit to a dendritic accumulator. The size of the effect that a synapse had on its target neurone was not directly related to the presence or absence of a spike in the axon. Instead, the model synapse contained a reservoir of leaky active neurotransmitter. Each incoming spike deposited neurotransmitter into the reservoir as shown in Fig. 14. The amount of activation passed to the target neurone at each time step was equal to the amount of active neurotransmitter, independent of the presence of a spike. The effect of this procedure was to spread the single-point spikes out in time in a more biologically plausible manner.

Fig. 14

Example neurotransmitter decay timecourse for a static synapse processing a simple spike train

The static synapse model was extended to a dynamic model by the addition of one or more facilitation and/or depression components. Pre-synaptic facilitation and post-synaptic depression were modelled using a generalised abstraction of residual calcium ion effects. This was based on a simplification of the Maass and Zador (1998) model.

Each facilitation or depression component comprised a calcium reservoir variable. Each reservoir was characterised by two parameters, which described the effect magnitude of a spike, and the calcium decay rate. If a spike was present at the synapse, then the magnitude value was added to the calcium reservoir. At all time steps the reservoir value decayed slowly, and exponentially, back to zero. The calcium reservoirs for each facilitation/depression component were independent.

Each spike arriving at the synapse deposited a quantity of neurotransmitter into the main reservoir equal to the sum of the facilitation/depression calcium reservoirs. Synapses with only a single component were restricted from changing the direction of their effect from that of the baseweight (i.e. excitatory synapses with a depression component could not become inhibitory, and vice versa).

The update function for a single dynamic synapse component at each time step is given as pseudo-code in Algorithm 3.

Algorithm 3 Dynamic synapse component pseudo-code

Parameters

Pre-synaptic axon (16 bit signed integer, source_firing) Equivalent to the axon pulse train variable of the source neurone.

Axonal transmission delay (unsigned integer range 1–16, delay) The delay in update cycles between a spike being created at the axon SIZ until it reached the synapse. The value refers to the bit number the source axon to which the synapse is connected.

Post-synaptic dendrite (16 bit signed integer, dest_synpot) Equivalent to the dendritic accumulator of the target neurone. In addition, synapse-on-synapse connections were modelled by setting this to be the calcium reservoir of the target synapse.

Active neurotransmitter decay rate (8 bit unsigned integer, decay) The amount of active neurotransmitter decayed by halving its magnitude this number of times at each time step. In all the reported work this parameter was held constant at one.

Base weight (16 bit signed integer, base_weight) Each spike passing through the synapse incremented the active neurotransmitter reservoir by this value.

Component magnitude (16 bit signed integer, compN_mag) In dynamic synapses, each spike incremented the calcium reservoir of facilitation/depression component N by this value. Thus positive values resulted in pre-synaptic facilitation and negative values in post-synaptic depression.

Component decay (8 bit unsigned integer, compN_tc) At each time step the magnitude of the reservoir was halved this number of times, and the resulting value subtracted from the current magnitude. A small value therefore produced a rapid decay rate and vice versa. Although somewhat convoluted, this calculation method was necessary in order to achieve real-time processing speed on the robot.

Variables

Active neurotransmitter (16 bit signed integer, active) The amount of active neurotransmitter present in the synaptic reservoir at a given timestep. If a spike was present at the synapse then neurotransmitter was deposited in the reservoir, according to

$$ {{\text{active}}(t) = {\text{active}}(t) + {\text{baseweight}} + {\text{compNweight}}(t)+\cdots } $$

(2)

The active neurotransmitter decayed asymptotically to zero according to

$$ {{\text{active}}{\left( {{\text{t}} + {\text{1}}} \right)} = \frac{{{\text{active}}(t)}} {{{\text{2}}^{{{\text{decay}}}} }}} $$

(3)

Activation was transmitted to the dendritic accumulator of the target neurone according to

$$ {{\text{destsynpot}}(t) = {\text{destsynpot}}(t) + {\text{active}}{\left( {{\text{t}} + {\text{1}}} \right)}} $$

(4)

Calcium reservoir size (16 bit signed integer, compNweight) The amount of calcium present in the reservoir of each facilitation or depression component at a given timestep. If a spike was present at the synapse then calcium was deposited in the reservoir, according to

$$ {{\text{compNweight}}(t) = {\text{compNweight}}(t) + {\text{compNmag}}} $$

(5)

The reservoir size decayed asymptotically to zero according to

$$ {{\text{compNweight}}{\left( {{\text{t}} + {\text{1}}} \right)} = {\text{compNweight}}(t) - \frac{{{\text{compNweight}}(t)}} {{{\text{2}}^{{\rm compNtc}} }}} $$

(6)