Delay-Induced Multistability and Loop Formation in Neuronal Networks with Spike-Timing-Dependent Plasticity

Abstract

Spike-timing-dependent plasticity (STDP) adjusts synaptic strengths according to the precise timing of pre- and postsynaptic spike pairs. Theoretical and computational studies have revealed that STDP may contribute to the emergence of a variety of structural and dynamical states in plastic neuronal populations. In this manuscript, we show that by incorporating dendritic and axonal propagation delays in recurrent networks of oscillatory neurons, the asymptotic connectivity displays multistability, where different structures emerge depending on the initial distribution of the synaptic strengths. In particular, we show that the standard deviation of the initial distribution of synaptic weights, besides its mean, determines the main properties of the emergent structural connectivity such as the mean final synaptic weight, the number of two-neuron loops and the symmetry of the final structure. We also show that the firing rates of the neurons affect the evolution of the network, and a more symmetric configuration of the synapses emerges at higher firing rates. We justify the network results based on a two-neuron framework and show how the results translate to large recurrent networks.

Multicoding in neural information transfer suggested by mathematical analysis of the frequency-dependent synaptic plasticity in vivo

“…However, they decay exponentially quickly, and thus although these two studies differ in how far past spikes are calculated, they may be both good approximate representations of the post-synaptic events. Also, the results of the STDP model taking into account dendritic and axonal propagation delays, reported by Madadi Asl et al do not contradict ours 26 , 27 . They showed that, in both two-neuron and network motifs, high-frequency firings promote bidirectional connections, indicating a large proportion of neurons with large synaptic weights.…”

“…Recently, Madadi Asl et al revealed that STDP model incorporating dendritic and axonal propagation delay can adequately explain the existence of recurrent connections between pairs of neurons in the cerebral cortex 26 . They found that the firing frequency plays an essential role in the formation of connectivity patterns in Two-Neuron Motif 27 . Moreover, firing variability, as well as the statistical properties of the spike frequency, seems essential for real-time information processing 28 .…”

Dendritic and Axonal Propagation Delays May Shape Neuronal Networks With Plastic Synapses

“…where A + ( A − ) and τ + (τ − ) are the learning rate and the effective time window of synaptic potentiation (depression), respectively, and sgn(Δ t ′) is the sign function. Δ t ′ = Δ t +ξ is the effective delayed time lag between pre- and postsynaptic spikes at the synaptic site (Madadi Asl et al, 2017, 2018a). Δ t = t post − t pre is the original time lag between pre- and postsynaptic spike pairs, and ξ = τ d − τ a is the difference between dendritic and axonal propagation delays.…”

“…By assuming that the neurons remain phase-locked, it was illustrated that the two-neuron results can be extended to recurrent networks of spiking neurons (Madadi Asl et al, 2017, 2018a). Different combinations of dendritic and axonal propagation delays can lead to the emergence of symmetric connections, i.e., either two-neuron bidirectional loops, in the case that dendritic propagation delays are greater than the axonal delays (Figure 2A), or loosely connected motifs when axonal propagation delays are greater than the dendritic delays (Figure 2C) (Madadi Asl et al, 2017).…”

“…Although this assumption simplifies the theoretical calculations and reduces the computational cost of multiscale computer simulations, it renders mathematical models unable to provide insight into relevant physiological mechanisms. However, a number of theoretical and computational studies have shown that propagation delays modify weight and neuronal dynamics by affecting the co-evolution of synaptic strengths and neuronal activity, and therefore, shaping the emergent functional and structural properties of plastic neuronal networks (Lubenov and Siapas, 2008; Aoki and Aoyagi, 2009; Kozloski and Cecchi, 2010; Rubinov et al, 2011; Knoblauch et al, 2012; Babadi and Abbott, 2013; Kerr et al, 2013; Madadi Asl et al, 2017, 2018a), where the synaptic strengths are regulated by spike-timing-dependent plasticity (STDP) (Gerstner et al, 1996; Markram et al, 1997; Bi and Poo, 1998; Song et al, 2000). Hence, incorporation of time delays in mathematical models can significantly modify the dynamical properties of neuronal systems, such as the emergence of different connectivity patterns (Madadi Asl et al, 2017, 2018a), affecting the dynamics of fixed points and synchronization properties between interconnected neurons (D'Huys et al, 2008; Popovych et al, 2011), and the emergence of different multistable dynamical attractors (Song et al, 2009; Madadi Asl et al, 2018a).…”