Structural inhomogeneities in synaptic efficacies have a strong impact on population response dynamics of cortical networks and are believed to play an important role in their functioning. However, little is known about how such inhomogeneities could evolve by means of synaptic plasticity. Here we present an adaptive model of a balanced neuronal network that combines two different types of plasticity, STDP and synaptic scaling. The plasticity rules yield both long-tailed distributions of synaptic weights and firing rates. Simultaneously, a highly connected subnetwork of driver neurons with strong synapses emerges. Coincident spiking activity of several driver cells can evoke population bursts and driver cells have similar dynamical properties as leader neurons found experimentally. Our model allows us to observe the delicate interplay between structural and dynamical properties of the emergent inhomogeneities. It is simple, robust to parameter changes and able to explain a multitude of different experimental findings in one basic network.
Distributions of synaptic weights are known to have a large influence on the dynamics and information-processing properties of neural circuits [1–5]. Recent electrophysiological studies have shown that the distributions of the amplitudes of excitatory postsynaptic potentials (EPSPs) in cortical [6,7] as well as hippocampal [5] networks are typically long-tailed and span several orders of magnitude. This characterizes a topological configuration in which the majority of synaptic connections are weak and a small number are much stronger [6–9]. At the same time, distributions of firing rates during spontaneous activity have been found to be long-tailed [10] and there is increasing evidence that long-tailed distributions play a fundamental role in brain functioning [11–13]. Studies of microcircuit connectivity have also demonstrated a number of significant inhomogeneities, as well as correlations amongst cell activity. Fine-scale functional subnetworks have been found in cortical networks [1,3, 14], and it has been shown that cells with strong outgoing synapses cluster together [3,7]. Both the clustering of the highly active cells [3] and the presence of strong synapses [1] are likely to play an important role for network dynamics and information processing in the neocortex. Apart from structural inhomogeneities in networks, the impact of individual neurons on the dynamics of neural networks may also differ substantially. A number of recent in vitro studies of 1D and 2D dissociated developing cortical and hippocampal cultures have shown that such networks typically express spontaneous neural activity characterized by network bursts, and ongoing repetitions of distinctive firing patterns within those bursts [15–19]. Furthermore, several studies have shown that the activity of certain neurons reliably precedes population spikes [16–19]. These early-to-fire neurons have been termed leader neurons [18] and have been found to form functionally connected networks, the activity of which collectively precedes most of the observed population bursts [19]. In the 1D case, population bursts have been found to be triggered by “burst initiation zones” [17] and in the 2D case recent studies [18,20] have shown that leader neurons not only precede but also are able to initiate population bursts. Nevertheless, the underlying network structure and specific topological properties of leader neurons and subnetworks of such cells remain to be discovered. Experimental studies in constrained [21] or chemically modulated [22] cultures give reason to believe that a complicated process of self-organization underlies their emergence. However, from the modeling point of view, little is understood about how strong inhomogeneities, such as the aforementioned, could evolve in a self-organized manner by means of activity-dependent synaptic plasticity. Synaptic plasticity is widely believed to be the basis for learning and memory and shapes the distribution of synaptic weights, which has been found to be typically long-tailed in a number of recent experimental studies [6,7]. The influence of long-tailed weight distributions on network dynamics has been studied in a number of recent works [2,4, 5,12,23], and it has been shown that such distributions can increase spike-based information transfer and facilitate information processing in neural networks [5,24]. Yet, also in the latter case little is known about how experimentally observed activity-modulated synaptic plasticity, such as spike-timing-dependent plasticity [25–27] (STDP) and homeostatic plasticity [28,29] could lead to a symmetry-breaking in the distributions of synaptic weights of an initially homogeneous network, or to the preservation of inhomogeneities prescribed by certain initial conditions [30,31]. Even less is known about how networks could self-organize to simultaneously express the aforementioned properties such as having both long-tailed distributions of weights and firing rates at the same time [2], cells with different dynamical effects on network activity, and subnetworks with distinct dynamical and structural properties similar to the ones observed experimentally. Although recent theoretical models have proposed mechanisms that lead to the emergence of long-tailed synaptic weight distributions [32,33], and in case of the SORN model [33] to other interesting aspects of self-organization, these either employ a specially tailored plasticity rule for this purpose [32], or do not express long-tailed firing-rate distributions [33]. It is well known that networks of spiking neurons can exhibit strongly irregular dynamics if excitatory and inhibitory inputs to each neuron balance, such that the network is driven by fluctuations in its input, resulting in each neuron producing spikes at irregular times [24,34,35]. Such networks are called balanced state networks. They combine computational simplicity and dynamics that closely resembles activity recorded electrophysiologically in vivo from cortical cells of behaving animals [36,37]. This makes balanced state networks a very attractive and widely used theoretical model for cortical tissue [34,35,38,39]. Our goal for this study was to investigate processes of self-organization in such networks brought about by means of synaptic plasticity. We therefore consider a random network of spiking neurons in the balanced state, operating in the asynchronous irregular (AI) regime [34,35] that is believed to be a good fit to the activity of cortical networks in vivo [36,37]. We endow it with two activity-dependent synaptic plasticity rules, namely spike-timing-dependent plasticity [26,40,41] (STDP) and synaptic scaling [29]. In its prototypical form, STDP causes long-term potentiation (LTP) of a synapse if presynaptic spikes repeatedly occur some milliseconds before postsynaptic ones, whereas a reversed temporal order causes long-term depression (LTD). Since its initial discovery at glutamatergic synapses [25,26,41], many forms of STDP have been observed experimentally [42], also such forms acting at GABAergic synaptic connections [43–47] and many models have been proposed [27,48–52] to describe the mechanisms and dynamics of STDP. In particular, STDP acting at inhibitory-excitatory connections has been shown to influence spiking dynamics of hippocampal pyramidal neurons [45]. Several recent modeling studies have shown that inhibitory STDP has a stabilizing effect on network dynamics [53–56], and others also have started addressing questions of functional aspects of inhibitory plasticity [57,58]. Another well-studied form of synaptic plasticity is synaptic scaling [29,59], a form of homeostatic synaptic plasticity [28] that describes the up- and down-regulation of a neuron’s synaptic input in order to keep a preferred mean target firing rate. It is well known that STDP alone can lead to instabilities in network dynamics due to effects of positive resonance which result in runaway excitation, and that endowing a random network solely with a multiplicative or power-law STDP rule acting at excitatory-excitatory synaptic connections while keeping the other synaptic efficacies fixed does not lead to stable effects of self-organization [30,31]. Combinations of STDP and synaptic scaling, however, are known to be able to keep network dynamics in a stable and biologically plausible regime [60], and to support non-trivial computations underlying many optimization tasks [61]. Furthermore, it has been shown that combining Hebbian and homeostatic plasticity rules both has a stabilizing effect on network dynamics [62], and it has been found to express structure-building properties in simple model networks [63] of rate-based neurons. Here, we investigate how the inclusion of synaptic scaling and inhibitory STDP could bring about self-organization in spiking networks. Our model network develops both long-tailed distributions of synaptic weights and firing rates, similar to those observed experimentally [6,10]. Moreover, a delicate interplay between dynamics and synaptic plasticity leads to the emergence of a special group of neurons that we call driver neurons. They form subnetworks that can take strong influence on network dynamics and share properties of certain neurons and subnetworks found experimentally [1,3, 18,19]. The phenomena that we observe are generic and hold under alternations of the plasticity rules and their parameters.
After a transient phase, the weight distributions of the dynamic E-E and I-E synaptic connections have settled to their new stable shapes (see Fig 1). The mean synaptic strength of E-E connections is kept fixed at a value of 1 by the synaptic scaling rule, but the variance grows rapidly (see Fig 1,1D). We observed that E-E connections undergo a symmetry breaking and that we obtain a long-tailed distribution of synaptic weights after convergence (see Fig 1,1B). The EPSP distributions found in cortical and hippocampal networks of excitatory neurons are typically long-tailed [5–7]. Such types of weight distributions can lead to optimal enhancement of the responses of individual neurons to input and are thus beneficial for information transmission at strong synapses, as was shown recently [4,5, 24]. We used a maximum likelihood estimator for the exponent to fit a power-law distribution with a cutoff into the middle part of the synaptic weights distribution, omitting the strongest 5% of the excitatory synaptic connections. Our fitting procedure was modified from [64]. We found a power-law exponent α = −1. 92 and an upper cutoff xmin = 0. 205, Fig 1B. Although visually the fit does look good, with the amount of data we produced, it is possible to reject the power-law hypothesis [64]. However, it is clear that the truncated distribution has a long-tail behavior and that the probability decays approximately as a power-law. We obtained similar results when we exchanged the additive STDP rule with a partly or fully multiplicative one (see Section Different STDP rules). The weights of the I-E connections evolve to a near Gaussian form. This is due to the fact that inhibitory STDP is subject to negative feedback and thus yields unimodal distributions of synaptic strengths even in the case of a purely additive plasticity rule [54]. In addition to expressing long-tailed weight distributions after convergence, the network rests in the AI regime and expresses approximately log-normal firing rate distributions throughout the transient state (see Fig 1A). We observe many cells firing at very low rates close to 0 Hz and only a few cells firing at rates up to 30 Hz, a property in line with experimental data obtained during spontaneous cortical activity in vivo [10] and ubiquitous in brain networks [12]. While log-normal rates are known to be a general and robust property of random balanced state networks with homogeneous weights [23], the combination of both long-tailed distributions of firing rates and synaptic weights is not a straightforward property [2]. Plasticity in the network leads to the development of few exceptionally strong excitatory synapses (see Fig 1B). Interestingly, many of these synapses are found on excitatory neurons which have predominantly strong outgoing connections and which fire at higher than average rates. As we will show in the following, the elevated firing rates are in fact causal for the emergence of their strong outgoing weights (see Section Emergence of driver neurons). We call these neurons driver cells (or driver neurons) and characterize them by distributions of outgoing excitatory synaptic weights with a high mean value (see Fig 2A). To define the group of driver neurons we take the top 0. 5 percent of the excitatory cells with the largest mean outgoing synaptic strength (see Fig 2A). As our network consists of 4000 neurons, this amounts to 20 driver cells in the network. As the distribution of mean outgoing weights in the network is unimodal, there is no clear-cut threshold separating any group of cells from the rest of the population in this distribution. The particular choice of the threshold results in a very strong dynamical impact of the driver neurons as discussed in section Dynamical impact of driver neurons. Another possibility to define driver neurons would be to use a threshold located at three standard deviations above the mean. In this way we select neurons that have much stronger outgoing weights than expected under the hypothesis of normally distributed mean outgoing strengths. In contrast to the case of a normal distribution where this choice will result in an expected 0. 23% of all cells, we classify 3% of the population as driver neurons using the aforementioned criterion. Using this choice, there is also a detectable dynamical difference between the driver neuron group and the rest of the network, though not as strong as for the 0. 5% threshold (see Section 9 in S1 Text). The clustering of strong outgoing synapses in the network is shown in Fig 3 (see also Fig U, top left in S1 Text). Here, we plotted the quantiles of the distribution of the mean synaptic strength per neuron in the original network. For comparison we also plotted quantiles of surrogate data obtained by shuffling the synaptic weights among all excitatory synapses. This operation destroys all correlations in synaptic weights introduced by the plasticity mechanisms, while leaving network topology and the overall distribution of synaptic strengths unchanged. As we can see in Fig 3, for the shuffled networks (we show the mean and the standard deviation of 100 shufflings, standard deviation very small and almost invisible in the plot), most cells have a mean outgoing weight of around 1, the mean excitatory weight in the network. The self-organization of driver cells is due to a delicate interplay of network dynamics and synaptic plasticity. For example, driver neurons are much less pronounced in a model network where we do not include inhibitory STDP (see Fig 2B). We will describe the process of their emergence in more detail in the following. Looking at driver cells in the equilibrium network and at the same cells early in the network evolution (that we call future driver cells), we found that they fire at rates much higher than the network average (network average approximately 5 Hz, driver group average approximately 25 Hz, see Fig 2A). In the following, we will show that this is the main reason for the emergence of their strong outgoing synaptic connections: STDP dynamics of the excitatory synaptic weights in our network can be seen as a random walk on the closed interval [0, wmax]. Here, the probability to increase the weight grows with increasing synaptic weight, and for weights above a certain threshold the average impact on synaptic weight of each presynaptic spike is positive. In a model without homeostatic plasticity we observed that once a synaptic weight reaches this threshold it converges to its maximum with high probability, with a velocity proportional to the firing rate of the neuron. As we included a postsynaptic homeostatic plasticity rule in our model at E-E synapses that constrains the total sum of weights onto each neuron, this led to a competition of all excitatory synapses converging onto a given excitatory postsynaptic cell over a limited pool of total synaptic efficacy. One characteristic of many STDP rules as well as the one that we include in our model is that synapses connecting a highly active presynaptic cell with a less active postsynaptic one (in terms of their mean firing rates) tend to undergo LTP [65,66]. Thus, outgoing synapses of driver cells that fire faster than the average cell have a higher probability to undergo LTP. Synapses from future driver cells are the ones to predominantly win that competition over available synaptic efficacy, diminishing the influence of other cells (see Fig 4B and Section 4 in S1 Text). This ultimately allows driver cells to emerge and to have strong influence on their postsynaptic networks. For our model we support this by analytical considerations (see Methods). We observe that the higher firing rates of (future) driver cells is due to reduced inhibitory currents those cells receive (see Fig C, left in S1 Text). Currents in our model are influenced by two variables, synaptic weights and presynaptic firing rates. We find that the reduced inhibitory currents that drivers receive are a result of two separate effects of local network topology: First of all, driver cells have a lower than average number of converging inhibitory synapses. Secondly, inhibitory cells which are presynaptic to driver cells have on average a higher number of converging inhibitory synapses than randomly selected inhibitory cells of the network. The latter results in lower than average firing rates of the inhibitory cells presynaptic to driver cells (see Fig D in S1 Text) and the combination of these two effects leads to a permanently reduced inhibitory drive to driver cells. As the included inhibitory STDP rule is subject to a form of self-regulatory dynamics with negative feed-back that becomes stronger with increasing synaptic weight [54], inhibitory plasticity cannot fully compensate this reduced inhibitory drive to driver cells by increasing inhibitory weights: On the one hand, inhibitory STDP in our model tries to compensate the high rate of the under-inhibited neurons by increasing the converging inhibitory weights to those cells. On the other hand, each inhibitory presynaptic spike delays the postsynaptic spike for an increasing period of time with increasing inhibitory synaptic weight, thus decreasing the positive contribution of the STDP rule. This results in a situation in which the converging inhibitory weights become stationary at a value below the maximal inhibitory weight, even in the case of high postsynaptic firing rates as seen in the case of driver neurons (see Fig 4). Altogether, we thus find that driver cells in our model are mainly determined by (local) network topology and that their emergence is due to an interplay of all three plasticity rules active in the network. In order to test the sensitivity of the observed effects to changes in the network size, we simulate networks of 10,000 and 20,000 cells and find that also in these we obtain qualitatively similar results (see Section 10 in S1 Text). In the following, we will investigate dynamical and topological properties of the group of driver cells and compare them to randomly sampled groups of non-driver cells of the same size. One question we wanted to answer is whether the emergence of driver neurons influences the dynamics of the network. To answer this question, we forced both the group of driver cells and a group of randomly selected non-driver cells to fire two consecutive spikes. We observed the network response in both cases. We achieved this by providing two brief pulses lasting 0. 5 ms of a very strong constant current input to the group of stimulated cells. Those two pulses were separated by a delay of 2 ms to allow all cells to leave their refractory periods after emitting the first spike. We furthermore only considered cases in which none of the stimulated cells was refractory prior to stimulation so that all cells of the stimulated groups fired exactly two spikes within 2+ϵ ms, where ϵ ≪ 1 ms is dependent on the membrane potential of the cell prior to the stimulation. We chose this stimulation protocol to imitate a bursting activity in the chosen subpopulation. To probe a baseline response of the network, we stimulated the same number of randomly selected non-driver cells with the same protocol. In the latter case, the network firing rate rises shortly due to the induced simultaneous firing of the stimulated neurons, but there are no lasting effects on network dynamics (see Fig 5B). On the other hand, the stimulation of the driver cell group results in a prolonged elevation of the firing rate similar to a population spike (see Fig 5A). To get a more precise picture, we observed the network dynamics subject to the condition that a number of driver neurons spontaneously fire in a synchronous way. We considered events in which a certain fraction of the driver cells all fire within a 1 ms time bin and average the excitatory population rates before and after this event, obtaining a synchrony triggered average (STA) curve (see Fig 6A). For comparison, we sampled a random group of non-driver cells of the same size and considered coincident spikes from its members (see Fig 6B). The STA curves for the random group are symmetric around the moment of synchronization. This indicates that the probability of finding a given number of neurons from this group firing coincidently within 1 ms is higher when the network rate is higher than usual, but this event has no effect on network dynamics. On the contrary, two synchronous spikes from the driver group are sufficient to result in a detectable and causal elevation of the population firing rate. This effect becomes more pronounced for larger groups of driver cells firing synchronously (see Fig 6A). Choosing the top 3% of cells with the highest mean outgoing weights as drivers predictably diminishes the absolute impact of synchronous driver spiking on network dynamics, but qualitatively the results remain unchanged (see Section 9 in S1 Text). We also performed the same averaging in a setting in which a selected number of driver and random non-driver neurons are forced to spike coincidently by providing a brief stimulation with a strong constant current input (see Fig 6C). The response of the network to the additional synchronous spiking of driver neurons is similar to the case of spontaneous synchrony. A prolonged phase of elevated activity is observed and the effect grows with the number of activated neurons. On the other hand, even after a stimulation of many non-driver neurons, activity rapidly returns to the base-line level and no prolonged change in the network dynamics can be observed. A recent series of experimental studies performed in vitro [18–20] in dissociated hippocampal and cortical cell cultures reported the existence of certain special neurons termed leader neurons. They were characterized by their firing activity stably preceding population bursts in the culture. Leader neurons were found in a wide variety of dissociated cultures obtained from both hippocampal and cortical cells from embryonic, newborn (< 24h) and juvenile (P16–17) rats [18] and under a wide variety of feeding protocols. It was found that synchronized firing of leader neurons increases the probability of the initiation of a population burst above chance level [18]. Furthermore, recent experimental work [20] shows that leader neurons do not just passively precede bursting activity in the cultures, but can actively trigger it. Whereas our model network stays in the asynchronous irregular activity regime that differs greatly from the synchronized bursting behavior of the hippocampal cultures, there are still periods of elevated activity, and driver neurons fire preceding them and can cause such events if many drivers are triggered to fire simultaneously, a property shared with leader neurons. There are further properties that the driver neurons in our model share with leader neurons, for example the tendency to form functional subnetworks [18,19] (see Section Stability and topology of the driver neuron subnetwork). Moreover, it was shown [16–18] for leader neurons that the property of early spiking during population bursts is very likely to be caused by synaptic input (reduced inhibition or increased excitation, or a combination of both) to those cells and not intrinsic cell properties (e. g. a reduced firing threshold), a finding we also made for the driver neurons in our model. It is an interesting open question to elucidate what will happen if the network is set to stay in the bursting regime throughout its development, that we consider for a follow up publication. Moreover, a recent experimental study [3] assesses the existence of subnetworks of highly active excitatory cells in the somatosensory cortex of juvenile mice, expressing both characteristics of leader neurons and driver neurons and thus giving further experimental support for a unification of the two concepts. In earlier studies of balanced state networks with a power-law or multiplicative STDP rule acting at excitatory-excitatory connections [30,31], strong synapses arising due to a temporary symmetry breaking were not stable and disappeared after some time. In our model the strong synapses diverging from the driver neurons remain strong over long periods of time. Consequently, the property of belonging to the group of driver cells is stable over long periods of network time. This is due to the fact that in our model the property of becoming a driver cell is mainly determined by local network topology. It is an interesting question what will happen in much larger networks. To test for the sensitivity of the results to network size, we simulated networks of 10,000 and 20,000 neurons. We found that increasing network size in the considered ranges did not significantly change the shape of the converged weight distributions or the clustering of strong outgoing weights (see Section 10 in S1 Text). We furthermore observed that driver cells form “rich club” subnetworks in which most synapses are strong. This can be explained by the fact that driver cells emerge in waves, recruiting cells from their postsynaptic networks. As the outgoing synaptic connections of future driver cells which fire at much higher than average rates undergo LTP, this leads to elevated excitatory currents in the cells postsynaptic to driver cells and to an increase in their firing rates. This, in turn, increases the chances of the postsynaptic cells of also becoming driver cells. This iterative process terminates at some point as the total available synaptic weight in the network is limited by a homeostatic rule and the fact that inhibitory STDP up-regulates converging inhibitory weights onto cells with elevated firing rates. To illustrate the effect, we simulated 1000 different networks and looked at the driver cells that emerged in those networks and their subnetworks. To measure the connectedness of the subgroups we studied the number of synaptic links within two groups of n = 20 neurons each, the group of driver neurons and a group of randomly sampled non-driver neurons. We found that the driver cell group has a significantly higher number of synaptic connections Cdriver compared to the number of synaptic connections Crand in the random group. The random group on average expressed Crand = 7. 35 ± 3. 30 (mean ± standard deviation) synaptic connections in their subnetwork. Not surprisingly, this is very close to the number of expected synaptic connections in a random network of 20 nodes with connection probability p = 0. 02 which is C0 = 20⋅19⋅0. 02 = 7. 6. In contrast, we found on average Cdriver = 12. 14 ± 2. 65 predominantly strong synaptic connections in the driver neurons subnetwork, an almost twofold increase compared to the random group. Similar relations were found in recent experimental studies of developing cortical networks. Stable subnetworks of more active cells were found to express a higher amount of connectedness [3]. Moreover, a tendency to higher mean EPSP amplitudes with lower variances within highly connected subnetworks was found [7]. The question remains whether there are other topological properties that distinguish driver neurons from the rest of the population, apart from the reduced inhibitory in-degrees. To answer this question we measured both excitatory and inhibitory in- and out-degrees throughout 1000 different networks. We found that the number of incoming and outgoing excitatory synaptic connections does not distinguish driver neurons from the rest of the network in our model, as we can already see in the example network (see Fig 7A). To asses the observed difference in inhibitory in-degrees more clearly, for each network simulation we extracted the driver group after convergence of the synaptic weights and took one random group of non-driver neurons of the same size. As expected from the definition, the distribution of outgoing excitatory synaptic weights allows to distinguish the two groups of neurons dramatically (see Fig 7C). The distributions of incoming inhibitory degrees are also easily separable across the two groups (see Fig 7D). To compensate the smaller incoming inhibitory degree, inhibitory STDP up-regulates inhibitory weights converging onto driver cells such that they receive on average much stronger incoming inhibitory weights (see Fig 7E). But as discussed earlier, they none the less receive reduced inhibitory currents when compared to the network average (see Section Inhibitory STDP). To verify our hypothesis that driver cells in our model are mainly distinguished by properties of local network topology, we performed simulations of 1000 networks with varying initial conditions but a fixed network topology. To induce statistical fluctuations, we stimulated the networks with Poisson noise and then looked at the probability (assessed via the relative frequency) of each cell to belong to the driver group after the synaptic weights have converged. We found that the driver neuron population remained mainly unchanged independently of initial conditions and that roughly 30 different neurons were found in the driver cell group across all trials (see Fig 7B). The source of the variation between the outcomes is in the random initialization and input fluctuations. Altogether, the property of belonging to the driver group is thus not solely but mainly dependent on network topology. Another interesting question is how large the variation in local network connectivity has to be in order to allow for the emergence of driver cells. To answer this question, we simulated a fully homogeneous network in which all cells have the same in-degree of both excitatory and inhibitory connections. In this case the weight distributions almost remained delta peaks, i. e. each synaptic weight stayed w ≈ 1 even subject to plasticity and no driver cells emerged (see Fig I, left in S1 Text). Interestingly, already a slight amount of under-inhibition suffices to allow for the emergence of driver neurons. We demonstrated this by taking a random group of 50 cells in the fully homogeneous network and selectively pruned 10% of the inhibitory synapses converging onto each cell of the group. We found that already this small change in homogeneity suffices to allow the group to become driver cells (see Fig I, right in S1 Text). What is the role of inhibitory plasticity in our model? Similar to recent theoretical studies [53–56], inhibitory STDP plays a stabilizing role in our network setup. Yet, inhibitory plasticity furthermore plays a crucial role in the emergence of driver neurons, as we will see in the following. In order to assess the effect of inhibitory STDP more closely, we simulated networks with static I-E synaptic connections of constant weight (see Section 7 in S1 Text). The magnitude of the constant inhibitory weights was selected to be equal to the mean of the equilibrium distribution of inhibitory weights in the same network with inhibitory STDP. Without inhibitory STDP, the network still exhibits mostly asynchronous irregular activity, but with a higher amount of oscillations and with some cells expressing firing rates of up to 100 Hz (see Fig F in S1 Text). Also, the excitatory population firing rate almost triples (with a mean rate of around 15 Hz) in this case, whereas the inhibitory population firing rate does not increase so drastically. The excitatory weight distribution is also similar to the case of a network including inhibitory STDP (see Fig G in S1 Text), but outgoing strong weights cluster much less on the subset of neurons that constitutes the driver cell group in the former case (see Fig 2B and Fig U in S1 Text), making their dynamical impact on network dynamics much smaller (see Fig H in S1 Text). We furthermore found that this result does not depend on the actual value of the fixed inhibitory weight, but that it can be observed even for very weak or strong fixed inhibitory connections. How can this be explained? The difference between plastic and static inhibitory connections lies in the selective nature of the Hebbian inhibitory STDP rule [44]. Namely that it increases the synaptic weights converging on cells with higher firing rates more strongly than the ones converging onto cells with lower firing rates (see Fig 4A). This is the mechanism underlying its stabilizing property. Without inhibitory STDP, slightly under-inhibited neurons (that could develop into driver neurons in the fully plastic network) fire with higher rates and by this alone increase the firing rate of their postsynaptic partners. This happens on time-scales that are much shorter than the ones of synaptic plasticity. Thus, neurons that are post-synaptic to under-inhibited cells attain higher firing rates and start competing with their presynaptic partners over the available pools of postsynaptic weights limited by the synaptic scaling rule (see Fig 4B). This process, although also yielding long-tailed distributions, prevents the strong clustering of strong outgoing synapses on single cells (see Fig 2B and Section 13 in S1 Text). The cells with the highest mean outgoing weight also have an impact on network dynamics in this case, but this is much less pronounced than in the case of a network including inhibitory STDP (see Fig H in S1 Text). Do the results depend on parameter tuning? Do the the results generalize to learning rules other than additive STDP? For this we will refer to the previously described model with additive excitatory and inhibitory STDP rules and a synaptic normalization at excitatory synapses as the base model and study variations of it.
In this paper we examined the self-organization of inhomogeneous synaptic strengths in balanced networks. Beyond the development of long-tailed weight and rate distributions, we observed a clustering of the strongest outgoing synapses on a few neurons that we call driver neurons. This clustering stays qualitatively the same for different modifications of the STDP rules, homeostatic regulations, and network topology. Our analytic results demonstrate how the network enhances small initial inhomogeneities by a combination of three plasticity rules: excitatory STDP, inhibitory STDP, and homeostatic plasticity. We furthermore showed that inhibitory STDP can serve not only the purpose of circuit stabilization, but also how it might be central for structure formation in networks.
The sub-threshold membrane potential of each LIF neuron obeys τ m d V d t = - (V - E L) + I e syn - I i syn, where τm = 20 ms is the membrane time constant, EL = −60 mV denotes a leak term and I e syn, I i syn denote excitatory and inhibitory synaptic currents, respectively. Whenever the membrane potential crosses a spiking threshold Vthres = −50 mV, an action potential is generated and the membrane potential is reset to the resting potential Vreset = −60 mV, where it remains clamped for a refractory period τref = 2 ms. The excitatory synaptic currents are given by I e syn = w e c e norm g e, where ge denotes the presynaptic spike train that is convolved with a synaptic kernel function, c e norm = 1 mV is a normalizing factor and we denotes the synaptic weight normalized to values [0, wmax] with w e max = 20, and an initial weight we = 1. The negative synaptic current I i syn is defined analogously with c i norm = − 9 mV, w i max = 5, and an initial synaptic weight wi = 1. Synaptic connections are current-based with exponential kernel functions τ e d g e d t = − g e and τ i d g i d t = − g i. Here, τe and τi denote excitatory and inhibitory synaptic time constants, respectively. They are chosen as τe = 5 ms and τi = 10 ms, in accordance with fast-acting excitatory and inhibitory neurotransmitters. Synaptic parameters are chosen so that effective EPSP and IPSP amplitudes are comparable with experimental data [88]. EPSP amplitudes take values between 0 mV and 2. 25 mV (corresponding to a synaptic weight of w e max = 20), with 0. 16 mV corresponding to an excitatory synapse with weight w = 1. IPSP amplitudes take values between 0 mV and −11. 23 mV (corresponding to a synaptic weight of w i max = 5), with −2. 25 mV corresponding to an inhibitory synapse with weight w = 1. We consider a random, balanced state network of leaky integrate and fire neurons consisting of N = 5000 cells of which 4000 are excitatory (E) and 1000 inhibitory (I). The network is fully recurrent with all four connection types E-E, E-I, I-E and I-I present. The probability of a synaptic connection between any two neurons is p = 0. 02, a value chosen as a compromise between the higher connection probabilities found for neighboring cortical neurons and the lower values for more distant cells [6]. Synaptic connections are current-based with an exponential decay and modeled to be in accordance with fast acting glutamatergic and GABAergic neurotransmitters. In order to ease simulation and analysis we restrict our model to have mono-synaptic connections between pairs of cells, aggregating possibly several synaptic contacts of one pair of cells into one postsynaptic potential (PSP). The E-E (STDP+homeostatic plasticity) and I-E (inhibitory STDP) synaptic connections are dynamic, whereas the I-E and I-I ones are static (see Fig A in S1 Text). STDP in our model is implemented in a standard on-line fashion with exponential kernels and all-to-all spike pairings so that the weight update for a synapse connecting a pre- and a postsynaptic cell is given by Δ w (w) = { A + (w) exp (- (t post - t pre) / τ +) if t post - t pre > 0 - A - (w) exp ( (t post - t pre) / τ -) if t post - t pre ≤ 0, where tpre and tpost denote pre- and postsynaptic spike times. For excitatory-excitatory synapses, we consider either additive (A+ (w) and A− (w) constant), partly multiplicative (A+ (w) constant) or fully multiplicative STDP rules, see S1 Text for a description of the different rules. For inhibitory-excitatory synapses we only consider additive rules as in this case additive and multiplicative rules are equivalent [54]. For both excitatory and inhibitory connections, time constants were set to τ+ = τ− = 20 ms. For additive excitatory STDP, the amplitudes of LTP and LTD were chosen as A+ = 10−3 and A− = 1. 05A+, respectively, resulting in a negative integral of the STDP window. For the case of partly multiplicative and multiplicative STDP at excitatory synapses we chose A+ = A− = 10−3. For inhibitory connections, we set A− = 10−3 and A+ = 4A− as motivated by experimental findings [44] and yielding a positive integral of the STDP window [54]. In the simulations, A+ and A− are multiplied with the respective maximal weight for additive rules to obtain their effective values. Like the excitatory STDP rule, inhibitory STDP [44] in our model is Hebbian, increasing the synaptic weight if the postsynaptic cell fires within τ+ ms after a presynaptic spike, and decreasing it when a presynaptic spike occurs within τ− ms after a postsynaptic spike. We verified that the results do not strongly depend on the learning rates of the STDP rules by varying them one order of magnitude into each direction. This influences the convergence speed of the synaptic weights to the equilibrium distribution, but not the shape of the distribution itself. Homeostatic plasticity is implemented in form of synaptic weight normalization acting at the postsynaptic site of excitatory-excitatory connections. The normalization rule is defined by w scaled (i): = 1 ∑ j w j in (i) deg EE in (i) w in (i), where for an excitatory cell i, win (i) denotes the vector of incoming excitatory weights with components w j in (i) and deg EE in (i) denotes the excitatory in-degree. In the simulations, the above normalization of weights is applied every 100 ms, replacing the weight vector at each cell with its normalized version. We note that for long simulation times synaptic normalization is equivalent to synaptic scaling, assuming equal mean rates of the presynaptic cells and slow time-scales of plasticity. Yet, we also performed simulations of the network with synaptic scaling mechanisms acting on a slower timescale according to the following rule, analyzed for rate-based models as in [62]: d w d t = - γ (ν - ν 0) w 2. Here, ν denotes the firing rate of the cell, ν0 a target rate and γ a learning rate. For the simulations we chose γ = 10−6 and ν0 = 0 Hz. A scaling step was performed each 50 ms during the simulation and firing rates were computed using a sliding window of length 100 s. We observed that these networks show qualitatively the same behavior as the ones with the synaptic normalization rule, see Section 12. 3 in S1 Text. At the start of the simulation, the membrane potentials of the neurons were drawn from a uniform distribution between Vrest and Vthres. Subsequently, the network was driven by a constant depolarization of each cell, sufficient to depolarize each cell by 11 mV. This input drove the network to the asynchronous irregular (AI) regime of activity with a mean population firing rate close to 5 Hz. We chose a constant depolarization as input since we wanted to study self-organization in the network brought about by its own dynamics, rather than some structure present in the input. Additionally, this case was previously studied in [71], where also some properties of the static network were assessed, such as the expression of asynchronous irregular spiking activity. The distribution of firing rates and interspike intervals (ISIs) in the network is long-tailed with few cells firing at rates up to 30 Hz and many at low rates below 0. 1 Hz. As expected, the mean value of the distribution of coefficients of variation of the ISIs is close to 1, indicating irregular spiking activity of the network. In an alternative setup, we tested a network in which each cell is stimulated by a Poisson spike train of the same mean intensity and noted that this setup results in qualitatively the same results. We started with a network in which initially all synaptic weights have a constant value of 1 for all four types of synaptic connections between excitatory and inhibitory cells and then activate the synaptic plasticity rules. We also simulated networks with initially Gaussian and uniform weight distributions and obtained qualitatively identical results. After a transient phase lasting around 5 hours of network activity, the weights stabilized to their new long-tailed distributions. We observed that during this transient phase the network rests in the asynchronous irregular regime of activity. No significant difference in the mean firing rate of the different populations and no apparent visual difference in raster plots before and after plasticity can be observed, see Fig B in S1 Text. We furthermore verified that the obtained results do not strongly depend on the learning rates chosen by systematically varying them around the chosen value, increasing and decreasing them by up to one order of magnitude. This change in parameters influenced the speed of convergence of the synaptic weights to their equilibrium distributions, but not the shape of the distributions itself, while at the same time taking no influence on network dynamics. We studied analytically tractable reduced models that enable us to calculate STDP weight updates in a mean-field fashion and that allow us to give an explanation of the observed processes of self-organization in the network.
It is widely believed that the structure of neuronal circuits plays a major role in brain functioning. Although the full synaptic connectivity for larger populations is not yet assessable even by current experimental techniques, available data show that neither synaptic strengths nor the number of synapses per neuron are homogeneously distributed. Several studies have found long-tailed distributions of synaptic weights with many weak and a few exceptionally strong synaptic connections, as well as strongly connected cells and subnetworks that may play a decisive role for data processing in neural circuits. Little is known about how inhomogeneities could arise in the developing brain and we hypothesize that there is a self-organizing principle behind their appearance. In this study we show how structural inhomogeneities can emerge by simple synaptic plasticity mechanisms from an initially homogeneous network. We perform numerical simulations and show analytically how a small imbalance in the initial structure is amplified by the synaptic plasticities and their interplay. Our network can simultaneously explain several experimental observations that were previously not linked.
Self-organization in Balanced State Networks by STDP and Homeostatic Plasticity