Emergence of connectivity motifs in networks of model neurons with short- and long-term plastic synapses

Abstract

Recent experimental data from the rodent cerebral cortex and olfactory bulb indicate that specific connectivity motifs are correlated with short-term dynamics of excitatory synaptic transmission. It was observed that neurons with short-term facilitating synapses form predominantly reciprocal pairwise connections, while neurons with short-term depressing synapses form predominantly unidirectional pairwise connections. The cause of these structural differences in excitatory synaptic microcircuits is unknown. We show that these connectivity motifs emerge in networks of model neurons, from the interactions between short-term synaptic dynamics (SD) and long-term spike-timing dependent plasticity (STDP). While the impact of STDP on SD was shown in simultaneous neuronal pair recordings in vitro, the mutual interactions between STDP and SD in large networks are still the subject of intense research. Our approach combines an SD phenomenological model with an STDP model that faithfully captures long-term plasticity dependence on both spike times and frequency. As a proof of concept, we first simulate and analyze recurrent networks of spiking neurons with random initial connection efficacies and where synapses are either all short-term facilitating or all depressing. For identical external inputs to the network, and as a direct consequence of internally generated activity, we find that networks with depressing synapses evolve unidirectional connectivity motifs, while networks with facilitating synapses evolve reciprocal connectivity motifs. We then show that the same results hold for heterogeneous networks, including both facilitating and depressing synapses. This does not contradict a recent theory that proposes that motifs are shaped by external inputs, but rather complements it by examining the role of both the external inputs and the internally generated network activity. Our study highlights the conditions under which SD-STDP might explain the correlation between facilitation and reciprocal connectivity motifs, as well as between depression and unidirectional motifs.

Figure 1. Emergence of connectivity motifs in a toy model network.

Unidirectional (reciprocal) strong excitatory connections are indicated (A) as dashed (continuous) line segments, representing the topology of the network (B). Each model neuron receives periodic spatially alternating depolarizing current pulses, strong enough to make it fire a single action potential. Synapses among connected neurons display (C) short-term facilitation of postsynaptic potential amplitudes. Spike-Timing Dependent Plasticity (STDP) leads to strengthened connections and results in a largely reciprocal topology (D). Modifying the short-term plasticity profile into depressing (F) leads to a largely unidirectional topology shaped by STDP(G). Distinct motifs of strong connections arise from short- and long-term plasticities, due to distinct firing patterns (compare E and H), under identical external stimulation and initial connections. Parameters: pA, and Methods.

Figure 2. Statistics of motif emergence in a toy model network.

When decoupled from recurrent interactions, an isolated model synapse undergoes long-term changes depending on pre- and postsynaptic spike timing (A) and pairing frequency (B). Above a critical frequency (grey shading), spike timing no longer matters and long-term potentiation of synaptic efficacy (LTP) prevails on long-term depression (LTD). Panels C, D: The simulations of Fig. 1 were repeated times, each time starting from a random initial topology. STDP progressively induced a persisting non-random reconfiguration of strong connections, quantified across time by a symmetry index (see Methods). Neurons connected only by short-term depressing synapses evolved strong unidirectional connections, corresponding to a low symmetry index. This is displayed in panel C, as an average across the simulations (left panel). Initial and final distributions of symmetry index values are also shown (right panel, grey and black histogram respectively). Neurons connected only by short-term facilitating synapses evolved instead strong bidirectional connections with high symmetry indexes (D).

Figure 3. Strong external inputs may drive neurons to high firing rate and induce reciprocal motif emergence even in depressing networks.

As in Fig. 2, the synaptic connectivity matrix of a network of ten neurons was randomly initialized and pruned (A, B, i.e., pruning is indicated by the “X” symbols). An external weak input, in addition to internally generated activity, contributes to the emergence of non-random unidirectional motifs, resulting in an asymmetric matrix W (C, D). However, if five units of the same network (grey circles) are externally stimulated above the STDP critical frequency, a non-random connectivity emerges, featuring reciprocal motifs and a symmetric connectivity submatrix (E, F; upper left corner, dashed rectangle). The values indicated above panels A, C, E represent the symmetry index and its significance.

Figure 4. Mean-field analysis of firing rate equilibria, in homogeneous networks without long-term plasticity.

The firing rate of homogeneous recurrent networks (A), including short-term facilitating synapses or depressing excitatory synapses, was studied by standard mean-field analysis. Average synaptic efficacies are indicated by . Excitatory and inhibitory external inputs are modeled by a single term, taking positive, zero or negative values. A zero value corresponds to balanced excitatory/inhibitory inputs, while a non-zero value corresponds to unbalanced excitatory/inhibitory inputs. The steady-state firing rate (i.e., , in a.u.) are the roots of the equation (see Text S1), whose graphical solution is provided (B–D), for facilitating (grey) or depressing (black) synapses, without long-term plasticity. Panel D is a zoomed view of C. (Un)stable firing rate equilibria are indicated by filled circles (squares). Networks with facilitating synapses fire at higher rates than networks with depressing synapses, as emphasized by the grey shading.

Figure 5. STDP key features for motif emergence.

The pair-based STDP model, with temporal window shown in panel A and matching exactly Fig. 2 A, exhibits a different frequency dependency (panel C) than the triplet-based STDP model (Fig. 2 B). Modifying the triplet-based STDP parameters to ad hoc invert its temporal window (e.g., as in anti-STDP, panel B, compare to panel A), yet leaves its frequency dependency and the LTD-reversal (grey shading) unchanged (panel D). Repeating the study of Fig. 2 with these two modified models, we find that (i) the pair-based STDP fails to account for motif emergence (panels E, G, compare to Fig. 2), while (ii) anti-STDP succeeds (panels F, H, compare to Fig. 2).

Figure 6. Results from numerical simulations of large recurrent networks of model neurons with short- and long-term plasticity.

Homogeneous and heterogeneous recurrent networks made of Integrate-and-Fire model neurons were numerically simulated, under identical conditions. Panel A shows the comparison of the emergence of weak or no connectivity pairs (indicated as “-, -”), of unidirectional strong connectivity pairs (“”, “D, -”), and of reciprocal strong connectivity pairs (“”, “D, D”) for a homogeneous network of neurons connected by depressing synapses: strong unidirectional depressing connections significantly outnumber reciprocal depressing ones. The fractions of emerged motifs (black) is significantly different than the null hypothesis (white) of random motifs occurrence. Panel B repeats this quantification for a homogeneous network with facilitating synapses: strong connections are found only on reciprocal connectivity pairs (“”, “F, F”) and all emerging motifs are non-random. Panel C repeats the same quantification for a heterogeneous network with both short-term facilitating and depressing synapses. Emerging motifs display highly non-random features and confirm that reciprocal facilitatory motifs (“”, “F, F”) outnumber unidirectional facilitatory motifs (“”, “F, -”), and that unidirectional depressing motifs (“”, “D,-”) outnumber reciprocal depressing motifs (“”, “D, D”). Panels D–F display the steady-state firing rate distributions, corresponding to homogeneous depressing, homogeneous facilitating, and heterogeneous networks respectively. The plots confirm that heterogeneity in connectivity motifs is accompanied by bimodal firing rates above and below the critical frequency, represented here by a grey shading.

Figure 7. Mean-field simulation of a heterogeneous network with short- and long-term plasticity.

The firing rate evolution of a heterogenous recurrent network, including both short-term facilitating and depressing synapses (A), was estimated by numerically solving the corresponding mean-field equations. The average synaptic efficacies among and across populations, indicated by , , , and , undergo long-term modification. Panels B, D show the long-term changes of an isolated synapse (decoupled from recurrent interactions) depending on pre- and postsynaptic spike timing (i.e., , ) and frequencies (i.e., , ). When and are varied independently, long-term potentiation (LTP) and depression (LTD) emerge as in associative Hebbian plasticity. This suggests that and will become significantly stronger than and (C) and that such a configuration will be retained indefinitely. This was confirmed by simulations (E–F) plotting the temporal evolution of the firing rates (black trace) and (grey trace), and of the mean synaptic efficacies (F). The heterogeneity occurs by separation of emerging firing rates (Fig. 4), as emphasized by the grey shading. Parameters: , msec, with initial conditions for the maximal synaptic efficacies , and . Please note that without loss of generality we fix .

Figure 8. Mean-field analysis of firing rate equilibria, in homogeneous networks with overlapping short-term synaptic properties and no long-term plasticity.

Panel A represents the sketch of a recurrent network where a clear segregation between subpopulations of depressing- or facilitating-only synapses does not occur. A neuron has a probability of connecting to its postsynaptic target by a depressing synapse, and of connecting to its target by a facilitating synapticapse. Panel B plots the location of the equilibria of the firing rate , under distinct external inputs conditions and for increasing values of . For the same parameters of Fig. 4, stable equilibria move as a function of , taking intermediate values between the two extreme cases, i.e., and ; compare to panels D–F of Figs. 4 .