Heterosynaptic plasticity rules induce small-world network topologies James McAllister, John Wade, Cian O’Donnell Intelligent Systems Research Centre, Ulster University Introduction Heterosynaptic plasticity is a form of ‘off-target’ synap- tic plasticity where unstimulated synapses change strength. Here we propose that one purpose of heterosynaptic plastic- ity is to encourage small-world connectivity [6, 7]. We com- pare different plasticity rules in abstract weighted graphs, finding that they yield distinct network architectures. Heterosynaptic plasticity Heterosynaptic plasticity is where synapses that were not directly activated undergo weight changes. Figure 1. Illustration of homosynaptic and heterosynaptic plasticity It can assume either a cooperative or competitive role in the alteration of synaptic weights (Figure 2). Cooperative Competitive Figure 2. Cooperative (left) and competitive (right) directions of weight change in heterosynaptic plasticity Heterosynaptic plasticity can operate locally on single den- drites at neighbouring spines, or across neurons and whole networks [4]. Method: Heterosynaptic plasticity network model Figure 3. Cases for weight updates in model Each edge is either (1) between two active nodes , (2) between one active and one inactive node, or (3) between two inactive nodes . Based on these cases, we define the update rules: R1 =  wi,jn+1 = wi,jn + η1(1 −wi,jn) Case (1) wi,jn+1 = η2wi,jn otherwise  R2 =    wi,jn+1 = wi,jn + γ1(1 −wi,jn) Case (1) wi,jn+1 = γ2wi,jn Case (2) wi,jn+1 = γ3wi,jn, Case (3)    R3 =    wi,jn+1 = wi,jn + κ1(1 −wi,jn) Case (1) wi,jn+1 = wi,jn + κ2(1 −wi,jn) Case (2) wi,jn+1 = κ3wi,jn, Case (3)    where R1 is a homosynaptic rule, and R2 and R3 are versions of competitive/cooperative heterosynaptic rules. ηi, γi and κi are learning parameters. We set η1 = γ1 = κ1 = 0.2 η2 = 1 −η1 = 0.8 η2 = γ3 = κ3 = 0.8 γ2 = 1 −γ1/2 = 0.9 κ2 = κ1/2 = 0.1 Method: Activity patterns We define the activity patterns of the network based on different Beta distributions (with varying parameters α, β). This determines the probability of each node being active. Network synchrony, firing rates, spike bursts and synaptic weights follow a log-normal distribution (Fig 4) [3], which we can partially replicate with beta distributions (Fig 5 blue line), as well as examine other less realistic activity patterns. Figure 4. Log-normal distribution of neural activity (Petersen & Berg, 2016) Figure 5. Example Beta distributions α = 1.5, 4, 4; β = 4, 1.5, 4 Method: Graph theory measures Weighted clustering coefficient: ˜C = Z 1 0 Ct dt where Ct = C(At) for At ij = 1 if wij ≥t and 0 otherwise. C(i) = |{ejk : vj, vk ∈Ni, ejk ∈E}| ki(ki −1) Average shortest path length (Dijkstra’s algorithm) L = 1 n(n −1) X i,j∈V i̸=j dij Small-world measure ↓ Small-world topologies A small-world network [7] has high degrees of clustering & low average shortest path length. σ = C/Crand L/Lrand > 1 Results Below is a simulation of how the network characteristics evolve under the homosynaptic (R1) and heterosynaptic (R2 & R3) plasticity rules over 100 timesteps. Figure 6. Graph theory measures for the three rules (N = 50 nodes) Given the probability distribution of activity, we can find a closed form solution for the weight matrix, which saves numerical simulations: R1: w∞ i,j ≈ η1pij 1 −(pij −pijη1 −pijη2 + η2) R2: w∞ i,j ≈ pijγ1 1 −(pij(1 −γ1 −2γ2 + γ3) + qij(γ2 −γ3) + γ3) R3: w∞ i,j ≈ (pij(κ1 −2κ2) + qijκ2) 1 −(pij(−1 −κ1 + 2κ2 + κ3) + qij(1 −κ2 −κ3) + κ3) where pi,j is the probability of node i and node j being active, and qi,j is the probability of node i or node j being active. Figure 7. Weight matrices for example network of 20 nodes Results continued Small-world measures across network sizes Figure 8. Small-world measures across different sizes of networks Beta distributions activity & small-world measures Figure 9. Average small-world measures across different activity distributions (N=100) Distribution of