1 Introduction
2 Methodology
2.1 The implemented spiking model
2.2 Comparison with the canonical integrate and fire model
Integrate and fire model (I&F) | Our model | |
---|---|---|
Membrane potential | The canonical integrate and fire [25] represents the evolution of the neurone membrane potential through the time derivative of the Law of Capacitance: \(I(t) = C_m {\frac{{\text{d}}V_m(t)}{{\text{d}}t}}\)
‘Integrate’ refers to the behaviour of the model when input currents are applied resulting in the increase of the membrane voltage until it reaches a set threshold which initiates a spike (fire event). The I&F model does not implement the decay of the membrane voltage towards its resting potential. Thus the membrane will keep a sub-threshold voltage indefinitely until new input currents make the membrane cross the firing threshold | The evolution of the membrane potential over time is described by the variable u. The behaviour of u(t) depends on: (1) the machine state at time t, (2) the applied currents from incoming spikes and (3) the membrane potential leakiness (see below) |
Leakiness | The decay or leakiness of the membrane potential is implemented as an extension of the I&F model: the leaky integrate-and fire Model (LI&F) recreates the dynamics of a neuron by means of a current I flowing through the parallel connection of a resistor with a capacitor in an electrical circuit [17, 19]. The current I splits in the resistor R and capacitor C, as follows: \(I(t) = I_R + I_C = \frac{u(t)}{R} + C \frac{{\text{d}}u}{{\text{d}}t}\)
where the voltage across the capacitor C is depicted with u and represents the neuron membrane potential. By introducing the membrane time constant \(T_{\text{m}} = RC\), the above equation can be rewritten as: \(T_{\text{m}} \frac{du}{dt} = -u(t) + RI(t)\)
with \(T_{\text{m}}\) quantifying the rate at which u decays to its resting potential | The decay of the membrane potential u is implemented through the decay() process by using two different functions (negative_leak_kernel and positive_leak_kernel) to describe the hyperpolarization and depolarization processes, respectively: If Rest_pot \(< u < \theta\) then \(u = u -\) negative_leak_kernel else If \(u <\) Rest_pot then \(u = u +\) positive_leak_kernel |
Rest_pot is the resting potential and \(\theta\) the firing threshold. In our model, both negative and positive kernels implement exponential decay functions | ||
Spike initiation | The mechanism of spike initiation is established through a threshold condition: \(u(t) = \theta\). Thus, if a given threshold \(\theta\) is reached at \(t = t^{(f)} ,\) then the neuron is said to fire a spike at time \(t^{(f)}\)
| Same as I&F Fixed firing threshold |
Action potential | The form of the generated action potential is not described explicitly in the LI&F model [17]. Following the fire event, the potential is reset: \(u_{\text{reset}}\)
\(< \theta\). Then, when \(t > t^{(f)}\) the dynamic behaviour continues as described by the membrane time constant \(T_{\text{m}}\)
| Same as I&F During the generation of action potential, the system initializes the absolute_refractory_period timer |
Refractoriness | The absolute refractory period is generally implemented by temporarily stopping the dynamics immediately after the threshold conditions have been reached. After the stop time, the membrane potential dynamics start again with \(u = u_{\text{reset}}\) where \(u_{\text{reset}} < \theta\)
| Same as I&F The state of the system remains as absolute_refractory as long as the absolute_refractory_period timer is still alive |
Synapses | Following the framework of the I&F model, given a neuron i, its total input current is defined as the sum of all its incoming current pulses:
\(T_i(t) = \sum _j w_{ij} \sum _f \alpha \left(t - t^{(f)}_j\right)\)
where \(\alpha (t-t^{(f)}_j)\) describes the time course from the presynaptic firing time t(f) at neuron j and the arrival time t at postsynaptic neuron i. \(W_{ij}\) represents the synaptic weight or efficacy between neuron j and the postsynaptic neuron i. The postsynaptic current generated by an incoming spike depends on the elicited change in the conductance of the postsynaptic membrane [19] | Similarly to I&F, the total input current is also expressed as:
\(T_i(t) = \sum _j w_{ij} \sum _f \alpha (t - t^{(f)}_j)\)
However, in contrast with the I&F framework, in our model the postsynaptic current only takes into account the efficacy \(W_{ij}\) of the synapses but not the conductance of the postsynaptic membrane |
2.3 The virtual brain for the virtual insect
2.3.1 The STDP learning rule
2.4 Anatomy of the virtual insect
2.4.1 The sensory system
2.4.2 The motor system
3 Results
3.1 The SNN Netlogo-engine
Number of insects | Average ticks per second (tps) |
---|---|
1 | 10,000 |
2 | 6800 |
3 | 4000 |
4 | 3200 |
3.2 The virtual insect
Symmetric LTP/LTD amplitude change | Number of ticks (iterations) before collision-free movement |
---|---|
0.01 | 19,000 |
0.02 | 15,000 |
0.03 | 9000 |
0.04 | 7000 |