Storage of Phase-Coded Patterns via STDP in Fully-Connected and Sparse Network: A Study of the Network Capacity

Abstract

We study the storage and retrieval of phase-coded patterns as stable dynamical attractors in recurrent neural networks, for both an analog and a integrate and fire spiking model. The synaptic strength is determined by a learning rule based on spike-time-dependent plasticity, with an asymmetric time window depending on the relative timing between pre and postsynaptic activity. We store multiple patterns and study the network capacity. For the analog model, we find that the network capacity scales linearly with the network size, and that both capacity and the oscillation frequency of the retrieval state depend on the asymmetry of the learning time window. In addition to fully connected networks, we study sparse networks, where each neuron is connected only to a small number z ≪ N of other neurons. Connections can be short range, between neighboring neurons placed on a regular lattice, or long range, between randomly chosen pairs of neurons. We find that a small fraction of long range connections is able to amplify the capacity of the network. This imply that a small-world-network topology is optimal, as a compromise between the cost of long range connections and the capacity increase. Also in the spiking integrate and fire model the crucial result of storing and retrieval of multiple phase-coded patterns is observed. The capacity of the fully-connected spiking network is investigated, together with the relation between oscillation frequency of retrieval state and window asymmetry.

Keywords: STDP; associative memory; phase-coding; replay; storage capacity.

Figure 1

(A) The learning window A(τ) used in the learning rule in Eq. (1) to model STDP. The window is the one introduced and motivated by Abarbanel et al. (2002), $A(\tau )=a_p{\text{e}}^{-\tau /T_p}-a_D{\text{e}}^{-\eta \tau /T_p}$ if τ > 0, $A(\tau )=a_p{\text{e}}^{\eta \tau /T_D}-a_D{\text{e}}^{\tau /T_D}$ if τ < 0, with the same parameters used in Abarbanel et al. (2002) to fit the experimental data of Bi and Poo (1998), a_p> = γ[1/T_p + η/T_D]⁻¹, a_D = γ[η/T_p + 1/T_D]⁻¹, with T_p = 10.2 ms, T_D = 28.6 ms, η = 4, γ = 42. Notably, this function satisfies the condition ${\displaystyle {\int }_{-\infty }^{\infty }}A(\tau )d\tau =0$, i.e. Ã(0) = 0. (B) The phase of the Fourier transform of A(τ) as a function of the frequency.

Figure 2

The activity of 10 randomly chosen neurons in a network of N = 3000 fully connected analog neurons, with P = 30 stored patterns. The learning rule is given by Eq. (4) with φ^* = 0.24π. Neurons are sorted by increasing phase ${\phi }_i^1$ of the first pattern, and shifted correspondingly on the vertical axis. (A) The first stored pattern, that is the activity of the network given by Eq. (2) used to encode the pattern in the learning mode, with frequency ω_μ/2π = 20 Hz. (B) The self-sustained dynamics of the network, when the initial condition is given by the first pattern $x_i^1(0)$. The retrieved the pattern has the same phase relationships of the encoded one. In this case the overlap is $\left|m^1\right|\simeq 0.22$, and the output frequency is in agreement with the analytical value tan(φ^*)/2πτ_m = 15 Hz. (C) Same as in (B), but with φ^* = 0.45π. Output frequency is in agreement with the analytical value tan(φ^*)/2πτ_m = 100 Hz.

Figure 3

Maximum capacity α_c = P_max/N of a network of N = 3000 (red) and N = 6000 (blue) fully connected analog neurons, as a function of the learning window asymmetry φ^*. The limit φ^* = 0 corresponds to a symmetric learning window (J_ij−J_ji) that is to output frequency $\overline{\omega }/2\pi$ that tends to zero. The limit φ^* = π/2 corresponds instead to a perfectly anti-symmetric learning window, and output frequency $\overline{\omega }/2\pi$ = ∞. The intermediate value φ^* ≃ 0.25π gives the best performance of the network.

Figure 4

(A) Maximum capacity α_c = P_max/N for a network with 24³ analog neurons, with z connections per neuron, with φ^* fixed to its optimal value 0.24π. The red curve corresponds to γ = 0, that is to a network with only short-range connections. The green one to γ = 0.3, and the blue one to γ = 1, that is to a random network where the finite dimensional topology is completely lost. (B) Maximum capacity α_c = P_max/N, for the same values of N and φ^* and for connectivity z/N = 0.11, as a function of γ, for the 24³ lattice (red) and for a two dimensional 118² lattice (blue). (C) Same data as in (B), but as a function of the clustering coefficient C.

Figure 5

Recall of a pattern by the spiking IF model, with N = 1000 neurons and φ^* = 0.24π. One pattern (P = 1) defined by the phases ${\phi }_i^{\mu }$ is stored in the network with the rule given by Eq. (4). Then a short train of M = 150 spikes, at times $t_i={\phi }_i^{\mu }/{\omega }_{\mu }$ with ω_μ/2π = 20 Hz, is induced on the neurons that have the M lowest phases ${\phi }_i^{\mu }$. This short train triggers the replay of the pattern by the network. Depending on the value of T, the phase-coded pattern is replayed with a different number of spikes per cycle. Note that changing T in our model may correspond to a change in the value of physical threshold or in the value of the parameter ηÃ(ω_μ) appearing in the synaptic connections J_ij learning rule. (A) Thirty neurons of the network are randomly chosen and sorted by the value of the phase ${\phi }_i^{\mu }$. Then the phases of the encoded pattern are shown, plotting on the x axis the times $({\phi }_i^{\mu }+2\pi n)/{\omega }_{\mu }$, and on the y axis the label of the neuron. (B) The replayed pattern with threshold T = 85 is shown, plotting on the x axis the times of the spikes, and on the y axis the label of the spiking neuron. Black dots represent externally induced spikes, while red dots represent spikes generated by the intrinsic dynamics of the network. (C) The replayed pattern with threshold T = 50. (D) The replayed pattern with threshold T = 35.

Figure 6

Example of selective retrieval of different patterns, in a network with N = 1000 IF neurons and P = 5 stored patterns. Here we choose the asymmetry parameter to be φ^* = −0.4π. Thirty neurons are chosen randomly, and then sorted by the value of ${\varphi }_i^1$ in (A–C), and sorted by the value of ${\phi }_i^2$ in (D–F). In (A) and (D) we show the phases of the first two stored patterns, plotting the times $({\phi }_i^{\mu }+2\pi n)/{\omega }_{\mu }$ respectively for μ = 1 and μ = 2. In (B) and (E) we show the generated dynamics when a short train of M = 100 spikes corresponding to the μ = 1 pattern is induced on the network, while in (C) and (F) the dynamics when the μ = 2 pattern is instead triggered. The network dynamics selectively replay the stored pattern, depending on the partial cue stimulation. Note that, because the parameter φ^* here is greater than in Figure 5, the frequency of the retrieved pattern here is greater.

Figure 7

(A) Maximum capacity α_c = P_max/N of a network of N = 3000 fully connected spiking IF neurons, as a function of the learning window asymmetry φ^*. The spontaneous dynamics induced by a short train of M = 300 spikes with ω_μ = 20 Hz and phases given by encoded pattern μ = 1 is considered, with threshold set to T = 200. (B) The output frequency of the replay, as a function of φ^*, when P = 1, T = 200, N = 3000.