Neuron and network model

3.2.1. Network structure

The model consists of neurons which are described by their membrane potential V_i, and connected by synapses of strengthw_ij, whereidenotes the postsynaptic andjthe presynaptic neuron. The neurons are organized in different networks (see Fig. 3.1): Synfire chainsconsist of L pools denoted by Pl which contain N neurons each. Each neuron in a pool Pl has a

C

Figure 3.1.: Left: Illustration of the model structure. A readout network M receives con-vergent connections to from different synfire chains such as C1 and C2. By the competition between the respective weights, w₁ and w₂, the network determines which chain is the optimally responds at a time interval represented by the output unit inM. Right: Raster plot showing the spikes in the readout network Mand selected pools from the chainsC1 andC2. Each dot corresponds to a spike. InC1, activity propagates faster and with smaller jitterσ_P compared toC2.

probability ofp_S to be connected to any neuron in the subsequent poolP_l+1 with strengthw_S p(w_ij) =

p_S forw_ij =w_S

1−pS forwij = 0 ∀i∈ Pl+1, j ∈ Pl. (3.1) If all neurons in pool Pl fire nearly synchronously with a small temporal jitter, this induces on average N wS inputs in each neuron in the subsequent pool Pl+1. Thus, the firing times from the preceding pool are averaged and the jitter is reduced in the firing times of poolPl+1. As each neuron in poolPl+1 in turn projects on average toN w_S neurons, the activity in pool Pl+2 will be even more synchronized. If all neurons in the chain are disturbed by synaptic noise, the temporal jitter will not decrease to zero, but converge to a near-synchronized fixed point where the effect of the connectivity and the noise are balanced [70, 37].

Apart from the synfire chains, there is areadout networkMconsisting ofM neurons with no connections among each other (w_ij = 0 ∀i, j∈ M), but which connections from the synfire chains. A poolP_l is connected to a readout neuroni∈M by the rule

p(w_ij) =

p_M forw_ij =w_S

1−p_M forw_ij = 0 ∀j∈ Pl. (3.2)

network parameters neuron and synapses synaptic plasticity

Table 3.1.: Values of all model parameters that are used unless otherwise stated.

The set of all neurons in a given synfire chain is denoted by Cα, as all the parameters are identical across chains except for α, which is defined below. The values of all parameters regarding network connectivity are listed in the left column of Table. 3.1.

3.2.2. Neuron model and synapses

The neurons are modeled as leaky integrate-and-fire units embedded in a stochastic back-ground network. While the membrane potential V_i stays below a threshold V_thr, the time evolution of V_i is given by

τdV_i The first term models the leakiness of the neuron, while the others describe its input. Without any input (I_noise=I_ext = 0,S_k= 0∀k),V_i relaxes exponentially to the resting potentialV_rest with time constantτ. The second term is a sum over all S_k spikes in all neuronskwhich the neuron is connected to. t^sp_kj denote the times of each of these spikes, where j the number of the spike. I_noise represents synaptic input from the stochastic background network, which is collapsed into an excitatory and an inhibitory part

Inoise=ǫ+N(λ+, λ+)−ǫ−N(λ−, λ+), (3.4) where ǫ₊, ǫ₋ > 0 and N(m, σ²) denotes a random variable with a Gaussian distribution with meanmand standard deviation σ. In this form, the Gaussians approximate two Poisson processes with rates λ₊ and λ−, respectively. Using the standard parameter values (see Ta-ble 3.1, middle column), ignoring the thresholdV_thr and without further input, the membrane potential converges to a mean of hVi=−46.4 mV and a standard deviation of σ_V = 1.4 mV.

Whenever the membrane potential V_i crosses the threshold V_thr from below, the neuron i fires a spike. V_i is then set to the reset potential V_reset and the current time t is included in the set of firing times of neuroniby increasing the number of spikes S_i by one (S_i →S_i+ 1)

and settingt^sp_{i Si} =t. A spike in neuronj influences the membrane potential V_i of all neurons i:w_ij 6= 0 it is connected to presynaptically. The time evolution of the induced PSP in neuron iis described by anα function

PSP(t) = t

where α is the rise time of the PSP. The synaptic weights w_ij in Eq. 3.3 are normalized by α to ensure that the total impact of a single spike on the postsynaptic membrane potential does not change with α. As mentioned before, different synfire chains denoted by Cα will differ only in this parameter. No additional synaptic delays are incorporated, soα is the only parameter that determines the time course of the PSP. Introducing a distribution of delays does not qualitatively change the results.

3.2.3. Synaptic plasticity

The connections from the chains to the readout neurons {w_ij : i ∈ M} are subject to two forms of synaptic plasticity: STDP [13] and homeostatic plasticity [172]. STDP is applied after each spike t^sp_ik in neuron i. For all spikest^sp_jl in neurons that are connected to neuroni {t^sp_jl : w_ij 6= 0 ∨ w_ji 6= 0}, the time t=t^sp_jk−t^sp_il is calculated and the respective weights are updated by adding

∆w=

( Apexp(−t/τp) if t >0

−A_dexp(−t/τ_d) if t <0 (3.6) Thus, the synapse between two neurons is strengthened if the presynaptic spike occurs earlier than the postsynaptic spike (t > 0), but if this order is reversed (t < 0), the synapse is weakened. This introduces the notion of causality into the learning rule.

Synaptic weights under control of the STDP learning rule either decays to zero or diverges, depending on the network’s activity [175]. One way to prevent this is to complement STDP by a homeostatic learning rule, which adjusts the synaptic weights such that the network achieves a certain mean firing rate a_goal. A simple model of this mechanism is given by a proportional-integral feedback controller [175]

This equation is applied to updatewafter each trial based on the actual mean firing rateaof all readout neurons. This seems more appropriate then using it every time step, as homeostatic plasticity is believed to act slowly [172]. Parameter value for both forms of plasticity are listed in Table 3.1, right column.