A toy, recurrent microcircuit model with weak background external inputs.

We first consider a simplified representation of an excitatory microcircuit: a network of adaptive exponential Integrate-and-Fire (IF) units. Neurons are connected to each other through excitatory synapses (Fig. 1 B), whose efficacy undergoes short- and long-term plasticity, according to widely studied phenomenological descriptions of short-term synaptic dynamics (SD) [6] and of long-term spike-timing dependent plasticity (STDP) [15] (see Methods). At low firing frequencies, this STDP model reproduces the common temporal correlation window, i.e., the long-lasting plastic change depends on whether the presynaptic neuron fired before the postsynaptic or not (Fig. 2 A). At moderately high firing frequencies, however, the same model captures Hebbian associative plasticity, in the sense that neurons that fire together wire together regardless of their firing timing (Fig. 2 B). This is in agreement with the experiments [22], where above 30–40 Hz LTP prevails on LTD, even when spike-timing per se would promote LTD, i.e., . Below that critical frequency, LTP or LTD instead reflects causal or anti-causal relationships between pre- and postsynaptic firing times, respectively [9]. Details on implementation and parameters used are described in Methods and in Table 1.

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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.

https://doi.org/10.1371/journal.pone.0084626.g001

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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).

https://doi.org/10.1371/journal.pone.0084626.g002

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Table 1. Parameters employed in the simulations: STDP parameters are as in the minimal all-to-all triplet model described in Pfister and Gerstner (2006); short-term depression and facilitation parameters as in (Wang et al., 2006); neuron parameters are as in (Clopath et al., 2010).

https://doi.org/10.1371/journal.pone.0084626.t001

In addition to internally generated synaptic inputs, neurons receive weak external inputs deterministically played back over and over, as a traveling wave of activity (Fig. 1 B). Such a background stimulation imposes spike-timing correlations (as in the temporal code of Clopath et al. (2010)) and should be regarded an oversimplified generic e.g., thalamic, input with temporally-correlated structure.

We define two microcircuits, identical for all aspects of neuronal properties, maximal synaptic efficacy, anatomical connectivity, and external inputs, with the exception of the SD properties of the synapses. Specifically, one microcircuit includes only short-term facilitating synapses (Fig. 1 C), while the other includes only short-term depressing synapses (Fig. 1 F). Synaptic maximal efficacies, which are initialized as uniformly distributed random numbers, slowly evolve during the simulation into largely non-random configurations of strong links among weaker connections (Figs. 1 D, G), via STDP. At the steady state, these configurations match an experimentally observed co-occurrence: reciprocal motifs emerge in cell pairs more often than unidirectional motifs, when synapses are short-term facilitating; the opposite occurs when synapses are depressing.

In our simulations, this is revealed both by direct inspection of the synaptic efficacy matrix (e.g., see Fig. 3) and by quantification via a connection symmetry index (see Methods). When takes values close to , almost all of the existing strong pairwise connections are reciprocal. On the other hand, for values of close to , unidirectional or very weak connection motifs prevail.

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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.

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These results, summarized in Fig. 1 for a sample microcircuit composed by ten neurons, have been confirmed over repeated simulations and analyzed in Fig. 2 . Over a slow timescale, which is controlled by the plasticity rate parameter of the STDP model, the interaction of SD and STDP leads to a very small degree of symmetry (, mean stdev), with high significance () for networks with depressing synaptic connections (Fig. 2 C). Under identical external inputs, SD and STDP lead to a large degree of symmetry (, mean stdev), with, again, high significance () in about of the networks involving facilitating synaptic connections (Fig. 2 D). In the remaining of the cases, the resulting symmetry value was in the range , suggesting a degree of variability as found in experiments [4], [13], where the discussed motif correlations are not observed of the time. In this simplified example, the variability is attributed to the sparse structural connectivity imposed a priori to result in an irregular firing regime (see Methods). Doubling the simulation time (not shown) led to a very minor reduction of the variability on , by less than and only for the facilitating synaptic connections, suggesting that stationarity had been already reached. The synaptic connectivity and firing activity configurations of our networks are stable by virtue of the strict boundaries imposed on synaptic efficacy values, as in most numerical implementations of STDP [24]–[26].

It is worthy to note that the microcircuits including short-term depressing synapses collectively fired at Hz (Fig. 1 H), reflecting the externally imposed firing activity, while the microcircuits employing short-term facilitating synapses fired at Hz (Fig. 1 E) and exhibited irregular firing patterns, which emerged by the recurrent network structure. We underline that the firing rate of the facilitating network was above the critical frequency that separates the “temporal” mode of STDP from its “Hebbian” mode (i.e., as LTD reverts to LTP, see Fig. 2 B). Therefore, for facilitating networks, LTP prevails and the temporal correlations of spike times are no longer important. Similarly, the depressing network fires below the critical frequency, and LTP or LTD are determined by the timing of the pre- and postsynaptic spikes. In the following sections, we discuss in detail the mechanism underlining motif formation.

Mean-field analysis of homogenous microcircuits.

To understand our simulation results, and in particular the difference of internally-generated activity in the two cases, we employ standard Wilson–Cowan firing rate description of neuronal population dynamics (see Eq. 12 [27]). This kind of analysis, whose full details are provided in Text S1, is by no means novel but to the best of our knowledge never reported before in the context of motif emergence.

We initially consider recurrent networks of excitatory neurons, connected by synapses whose average efficacy is not a constant but changes on the short term as a function of the presynaptic firing rate, facilitating or depressing (see Fig. 4 A and Eq. 13). For the sake of simplicity, we mimic the effect of feedforward inhibition by an average extra input to the population: we study the case in which this input is zero, i.e., referred to as balanced inputs ( in Eq. 12), or in which it is set to a positive value, i.e., referred to as unbalanced inputs, ( in Eq. 12). Including recurrent inhibition does not qualitatively alter the validity of our conclusions (see Text S1, section 1.4).

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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.

https://doi.org/10.1371/journal.pone.0084626.g004

The firing rate of the population can be then studied via standard methods of dynamical system theory [28], i.e., analyzing the system of mean-field equations for SD and for neuronal dynamics, and evaluating the stability of their steady-state solutions.

While for simplicity we did not choose the parameters of the rate models to exactly match the IF simulations [29], [30], we are nevertheless able to conclude that homogeneous neuronal populations with short-term facilitating synapses generally fire at higher rates than populations with depressing synapses, for the same values of maximum synaptic coupling and external inputs , and when engaged in internally-generated reverberating activity [29]. In particular, firing rates are limited by an upper bound that inversely depends on the time constant for recovery from short-term depression ( – see Text S1, Eq. S23):(1)

It follows that, with other parameters being equal, a network with small values of , i.e., where the time scale of recovery from depression is very long, would fire slower than a network with comparatively larger values of , i.e., where the time scale of recovery from depression is very fast or negligible. These two cases correspond to the numerical parameters employed for the networks with depressing and facilitating synapses, respectively (see Table 1 and [4], [13]). For a detailed analysis see section 1.3 of Text S1.

Panels B and C in Fig. 4 illustrate graphically in the state plane output firing rate versus average neuronal input (i.e., ), i.e., the actual location of the population equilibria for populations with facilitating or depressing synapses. The stability of each equilibrium point is indicated by different marker symbols: circles for stable, squares for unstable equilibrium points. Similarly to Fig. 2 B, the approximate location of the STDP critical frequency has been indicated by a grey shading.

A simple mechanism for the emergence of motifs.

We have now reviewed that networks with short-term facilitating synapses fire on the average at higher frequencies that depressing networks, for the same external inputs and maximal synaptic efficacies. According to the STDP model implemented here, with parameters as in Table 1, the long-term average change of the maximal synaptic efficacy can be written in a concise form (see [15] and Eq. 16):(2)under the assumption that presynaptic and postsynaptic spike trains have Poisson statistics, as hypothesized in the previous paragraph. In Eq. 2, is the critical firing frequency threshold of the postsynaptic neurons, above which LTP occurs and below which LTD takes place. If is between the stable equilibria for the population firing rate of the network with depressing synapses and those of the network with facilitating synapses (as in our setup), it follows that the change of maximal synaptic efficacy in the “facilitating” network will be positive while in the “depressing” network will be negative. The maximal synaptic efficacies in the facilitating network will then continuously increase until reaching their upper bound, leading to bidirectional connectivity motifs. On the contrary, the maximal synaptic efficacies in the depressing network will decrease, leading to connectivity motifs that would depend on the spike-timing information only, e.g., as those imposed by the (weak) external background stimulus. At this point, the correlational “pre-post” temporal mode of the STDP model comes into play, enforcing unidirectional connectivity motifs to the “depressing” network. Indeed, causality in the spike-timing unavoidably leads to unidirectional reinforcing of either one of connection on a synaptic pair, but never on both simultaneously [14].

In Text S1, we show analytically that under the external stimulus STDP promotes unidirectional connectivity that reflects the asymmetric temporal structure of the inputs, following the analysis of Clopath et al. (2010).

Positive and negative controls.

Our analysis indicates that the motif formation is determined by the firing frequency of the neurons and by the features of the long-term plasticity, i.e., sensitive to both the timing and frequency of spiking activity. Here, we demonstrate these two points (i) by imposing a strong background external input, as in the rate-coding of [14], and (ii) by ad hoc altering the physiological dependence of the STDP model on spike timing or on frequency.

We first consider a depressing network with anatomical connectivity and synaptic efficacies randomly initialized as in Fig. 1 (see the Methods section). Figures 3 A, B shows the initial network structure and connectivity. After the exposure to a weak background external input, as in the temporal coding of [14] (i.e., as in Fig. 1), the network evolves only unidirectional motifs, resulting in an asymmetric synaptic efficacy matrix , see Figs. 3 C, D. However, if five units of the same network (grey circles) are externally stimulated so that they fire above the STDP critical frequency , a non-random connectivity emerges, featuring both unidirectional and reciprocal motifs and a symmetric connectivity submatrix (Figs. 3 E, F; upper left corner, dashed rectangle). This demonstrates that external activity can also impose connectivity motifs to the network, as in [14]).

We then confirm the minimal set of long-term synaptic plasticity features sufficient for motif emergence, by performing additional negative and positive control simulations (Fig. 5). We consider the pair-based STDP model [24], [31] and adjust its parameters to get an identical spike-timing dependency to the one predicted by the STDP model we used (i.e., compare Fig. 5 A to Fig. 2 A). Under these conditions, the resulting spike-frequency dependency is completely different (i.e., compare Fig. 5 C to Fig. 2 B), showing no reversal of LTD into LTP at high frequencies.

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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).

https://doi.org/10.1371/journal.pone.0084626.g005

On the contrary, by making minimal changes to the equations and parameters of the STDP model we used (see Methods and Supplementary Information, Text S1), it is possible to ad hoc reverse the temporal dependency of STDP, e.g., as observed experimentally for anti-STDP (aSTDP) [32], while leaving the spike-frequency dependence roughly intact (i.e., compare Figs. 5 B, D to Figs. 2 A, B). This case still features the reversal of LTD into LTP at high frequencies. We note that this modified aSTDP “triplet rule” serves only as a positive control and it is not meant to provide an accurate description of aSTDP, since more data are needed to access its firing rate dependence.

Panels E–H of Fig. 5 repeat the simulations of Fig. 2, and demonstrate that only when the realistic frequency dependence of STDP [22] is present, the heterogeneity in the network firing rates leads to the emergence of asymmetric connectivity motifs (compare Figs. 5 E, G or Figs. 5 F, H to Figs. 2 C, D). We therefore conclude that all long-term plasticity models that capture the temporal nature of the STDP at low frequencies and the LTD to LTP reversal at high frequencies (eg. [14], [33]) would also reproduce our results.

Large microscopic simulations of homogenous networks.

We further confirm our results by means of numerical simulations of larger recurrent networks, composed of IF neurons.

In order to increase the biological realism, in these simulations we also introduced fluctuating random inputs to each neuron, mimicking irregular background synaptic activity (see [34], [35] and the Methods). Each neuron thus receives an uncorrelated noisy current, as well as a periodic wave-like stimuli as in the toy model. As in Fig. 2, SD and STDP shape the maximal synaptic efficacies so that unidirectional depressing connections significantly outnumber the reciprocal depressing connections, while facilitating reciprocal connections prevail on unidirectional facilitating connections. Figures 6 A, B display the count of the occurrence of unidirectional versus reciprocal connectivity motifs.

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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.

https://doi.org/10.1371/journal.pone.0084626.g006

Figures 6 D–E show the heterogeneous distributions of the firing rates of the two networks: for the same external input, networks of model neurons with homogeneous depressing short-term plastic synapses fire at low rates, while networks with homogeneous facilitating synapses fire at higher rates. The symmetry index , computed after a very long simulation run, results in a value of for the depressing network and of for the facilitatory network.

Spike-timing and associative Hebbian plasticity in heterogenous networks.

We first examine the impact of STDP in a simplified two-neuron system, with one neuron projecting to the other via a single synapse. Figures 7 B, D show the long-term change in PSP amplitude as a function of the pre- and postsynaptic firing rates, at that single synapse. Since in a heterogeneous network the mean pre- and postsynaptic firing frequencies may differ from each other, we swept the firing frequencies of the two neurons throughout all the possible combinations, within a realistic range (i.e., 0–70 Hz). We studied two cases: each presynaptic spike precedes the postsynaptic spike by msec, i.e., , or vice versa, i.e., . We emphasize that only synapses established within neuronal pairs that belong to distinct subpopulations can experience heterogeneous pre- and postsynaptic firing rates. In this case, however, the impact of spike-timing information becomes negligible, as soon as pre- and postsynaptic neurons fire at different incommensurable frequencies. In the small minority of cases where this is not true, pre- and postsynaptic frequencies are (sub)multiples of each other, and a transient synchronization of spike times occurs periodically. In these circumstances, the timing information has a specific impact, as revealed graphically by the bright or dark straight lines in the plots of Figs. 7 B, D. In all other cases, the overall plasticity profiles reflect the conventional associative Hebbian LTP/LTD and its consequences [30], [36], [37]. This illustration captures the essence of Eq. 2, i.e., the firing frequency of the postsynaptic neuron determines whether the synapse will be potentiated or depressed. Therefore, connections to neurons firing with high frequencies will be strengthened, while connections to neurons with low firing frequencies will be weakened.

Mean-field analysis of heterogenous microcircuits.

Intuitively, the heterogeneous population of Fig. 7 A can lead to the emergence of realistic connectivity motifs. To illustrate this point, we first ignore that subpopulations might interfere with each other's firing rates: the facilitating and depressing subnetworks are still characterized by higher or lower firing rates, respectively, as previously presented for homogeneous networks. Then, the LTP/LTD maps of Figs. 7 B, D suggest that if such an initial asymmetry of firing rates emerges then it will be maintained indefinitely. The connections will be in fact increasingly weakened while the connections strengthen. The resulting configuration, sketched in Fig. 7 C, is thus stable. We tested and confirmed this statement under the mean-field hypothesis, by numerically simulating the full dynamics of Eqs. 12, 14, and 15, in addition to computing their equilibria. Figures 7 E, F display the mean firing rates of each subnetwork, and the time course of the mean synaptic efficacies. We remind the reader that the maximal synaptic efficacies act on the mean synaptic efficacies as scaling factors, hence a weaker value of corresponds to a weaker value of . However, the desirable configuration emerges only when the maximal efficacies across populations, i.e., and , are initialized to slightly weaker values than the maximum intra-population efficacies, i.e., and . This prevents the facilitating subpopulation to transiently, but irreversibly, drive above the activity of the depressing subpopulation. In Text S1, we partially relax this condition showing how weak synaptic connections across subpopulations may still evolve from fully homogeneous initial couplings.

Large microscopic simulations of heterogenous networks.

We further evaluate our results by numerical simulations of a large heterogenous network, as in Fig. 6 C (see Methods). These simulations involve Integrate-and-Fire units, subdivided in two subpopulations of equal size, with a structural connectivity set to approximately of all possible connections, as in the homogenous case.

As in the mean-field model, the scaling factors of the maximal synaptic efficacy across populations are initialized to weaker values than the intra-population terms. Each neuron receives an uncorrelated background noisy current as well as periodic wave-like stimuli, similar to the homogeneous case. As indicated by Fig. 6 F, the firing rate distributions is bimodal: neurons in the subnetwork of depressing short-term plastic synapses fire at generally low rates, while neurons in the subnetwork of facilitating synapses fire at higher rates. The location of the critical firing frequency for the STDP is represented again as a grey shaded area.

Results in Fig. 6 C show all the possible synaptic combinations. Qualitatively similar to the data of Wang et al. (2006), reciprocal motifs are significantly co-expressed with facilitatory synapses and unidirectional motifs with depressing synapses. The actual motif counts are compared to the null hypothesis of having statistical independence between the connection occurrence within a pair of neurons, estimated at a confidence interval upon the same hypothesis of Bernoulli repeated, independent, elementary events. The frequency of observing a connection between two neurons, regardless of its SD properties, is first estimated by direct inspection of the connectivity matrix . Then the conditional occurrence frequencies of a facilitatory synapse and of a depressing synapse are computed, given that a connection exist between two neurons. The null hypothesis for each possible combination is given by standard probability calculus, under the hypothesis of independence of identical events. For example, the occurrence frequency of observing by chance no connections within a neuronal pair is , while the occurrence of observing by chance a reciprocal motifs with mixed depressing and facilitating properties is .

Finally, as the symmetry index was computed over a very long simulation run, it resulted in a value of for the depressing subnetwork and of for the facilitation subnetwork.

Toy Network.

(Fig. 1 B, 10 neurons) consists of a nA constant current, as well as periodically repeating nA square pulses. Pulses occur as in a traveling wave of activity, which moves every msec from one unit, e.g., the th neuron, to its neighbour, i.e., the th neuron. In space, each pulse is delivered in turn to all neurons as an extremely narrow bell-shaped profile, with unitary peak amplitude and standard deviation of , resulting in neighbouring neurons being only weakly stimulated simultaneously. Each pulse is of sufficient amplitude to elicit firing in the unit where the bell-shaped profile is centred on, e.g., the th unit.

Large Network.

(Fig. 6, 1000 neurons) is as in the toy network with the addition of a (spatially) uncorrelated gaussian noisy term [29], with mean , standard deviation pA, and autocorrelation time length msec. Parameter is drawn randomly for each neuron of the network, before launching the simulation, using a normal distribution with mean pA and unitary coefficient of variation. The noisy current mimics asynchronous synaptic inputs from (not explicitly modeled) background populations [35], [63].