Self-organization in Balanced State Networks by STDP and Homeostatic Plasticity

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.

Each cell receives input resulting in a constant depolarization by 11 mV. This drives the network to the asynchronous irregular (AI) state, cf. [1]. The network displays long-tailed firing rate distributions with a mean firing rate of around 5 Hz (see Fig. B 3 STDP in the model STDP in our model is implemented in an on-line fashion as a pair-wise rule with all-to-all pairing. Its implementation is as follows. Denote by x and y the traces of spikes of a given pair of a pre-and a postsynaptic cell, respectively. Whenever a pre-or postsynaptic spike occurs, x and y are increased by quantities a + (x) and a − (y) and decay as specified by time constants τ + and τ − : Here, δ is the δ-function and t i and t j denote times of pre-and postsynaptic spikes. An STDP rule altering the synaptic weight w of a synapse connecting the pre-and the postsynaptic cell can then be written as where A + (w) and A − (w) are functions describing the magnitudes of synaptic depression and facilitation, respectively.
STDP rules for which the magnitude of A + (w) and A − (w) are independent of the weight w are called additive and can be written as where w min = 0 and w max denote chosen minimal and maximal synaptic weights and where Θ is the Heaviside step function.

Interplay of excitatory STDP and synaptic scaling
In order to understand the dynamics of the excitatory STDP rule and its interplay with the included homeostatic plasticity that leads to the emergence of driver neurons in our base model, we consider a reduced model in which 80 presynaptic neurons are connected to one postsynaptic neuron (this is the mean number of incoming excitatory connections in our base model). The synapses are as in the base model and subject to additive STDP (see Section 3). The postsynaptic neuron implements a synaptic normalization as described in the main text.
We observe that in a setting where all cells receive Poisson input of the same intensity such that they spike irregularly at rates of around 5 Hz, a random subset of the synaptic weights converges to the maximum while most synaptic weights remain small (data not shown).
In a setting where a number of presynaptic cells receive stronger input and fire at higher rates of around 25 Hz (close to the rates of driver cells in the base model) we observe that these cells develop strong outgoing synapses converging to the maximal synaptic weight in almost all cases (see Fig. 4). The faster firing cells win the competition over the limited total amount of synaptic weight available on the postsynaptic site (due to the weight normalization). This in turn is due to a fact that strong enough synapses are on average growing with each presynaptic spike, and thus neurons that fire more will get their synapses more potentiated in the same interval of time. For analytical results on this phenomenon see Methods in main text.

Inhibitory currents on driver cells
Driver cells in our model receive lower inhibitory currents (see Fig. C). This is a result of a lower inhibitory in-degree of those cells as well as reduced firing rates of the presynaptic inhibitory cells when compared to the network baseline (see Fig. D).

Analytical approximation using a Fokker-Planck formalism
In this section we mainly follow the lines of Nicolas Brunel's computations [3,4]. The steady state membrane potential distribution of a leaky integrate-and-fire (LIF) neuron driven by Poisson input modeled as delta pulses is given by [4] where Θ denotes the Heaviside step function, τ m denotes the membrane time constant, V th the threshold potential, V r the reset potential and µ = C e Jτ ν ext and σ = √ µJ the mean and variance of the external input (C e denotes the number of external connections, J the weight of the external connections and ν ext the mean external firing rate) to each neuron and ν its mean firing frequency defined via the consistency equation that yields for the mean firing rate In this setting, let us assume that two cells that are synaptically coupled via an excitatory delta synapse with amplitude w. The probability of a presynaptic spike to emit a postsynaptic one is then given by assuming stationarity in the firing rates of both cells and that the effect of the presynaptic spikes does not take strong influence on the firing rate and membrane potential distribution P (V ) of the postsynaptic cell. See Figure E(right) for an example of the distribution of P spike (w) assuming a distribution of V according to Figure E(left) with respect to a synaptic weight w of the connecting synapse.
Does Eq. 4 suffice to explain the emergence of driver neurons? The answer is no and this is due to the negativity of the integral of the STDP curve: Although each presynaptic spike at a synapse with weight w > 0 causes the postsynaptic neuron to cross threshold and spike with non-zero probability, this can happen at any instant and taking the integral over time still yields an LTD-dominated change in synaptic weight. An analytical model that seeks to explain the emergence of driver neurons has to consider the shift in spike times of postsynaptic neurons caused by presynaptic spikes. We developed such a model in the main text, see Methods.

Network without inhibitory STDP
Network dynamics in our model are highly sensitive to over-excitation due to increases in excitatory synaptic weights. Without inhibitory STDP, the population firing rate of the excitatory population quickly decouples from the inhibitory one and the whole excitatory population fires at elevated rates (see Fig. F). In particular, excitatory cells postsynaptic to future driver cells (i.e. cells that receive less than average inhibitory currents and thus fire at higher than average rates) receive a higher excitatory drive due to the increasing excitatory weights converging onto those cells and the high firing rates of driver cells. This results in growing excitatory weights of those cells. As a consequence, their synapses compete with synapses of driver cells for the available postsynaptic weight. As a result, only a fraction of the synapses diverging from possible future driver cells can converge to their maximal values, resulting in the cells not becoming driver cells. Without inhibitory STDP, excitatory weight distributions are still long-tailed but strong excitatory weights in the network are more homogeneously distributed rather than being clustered on driver cells. This situation is characterized by many excitatory cells having some strong outgoing synapses (see Fig. G). In the base model the results stay qualitatively the same if the learning rates of the inhibitory STDP rule are altered. We tested values between η + = 3η − and η + = 5η − . We observe that with a decreasing quotient of η − /η + the spread of the excitatory weight distribution also decreases.

Homogeneous networks and under-inhibition
In order to show that inhomogeneities in network connectivity are causal to the emergence of driver cells, we simulated a fully homogeneous network in which all cells have the same indegrees. In this case, the weight distributions are almost delta peaks, i.e. each synaptic weight remains w ≈ 1 even subject to plasticity and no driver cells emerge, see Fig. I(left).
Interestingly, already a slight amount of under-inhibition suffices to allow for the emergence of driver neurons. We demonstrate this by selecting a group of 50 cells in the fully homogeneous network and selectively pruning 10% of the inhibitory synapses converging onto each cell of the group. This small change in homogeneity suffices to allow the group to become driver cells, see

Varying the driver threshold
As discussed in the main text, we can vary the threshold for being a driver cell a bit and still get qualitatively similar results regarding the dynamical impact of driver cell spiking on network activity. To assess this, instead of selecting as driver cells the top 0.5% (amounting to n = 20) of the cells with the strongest outgoing mean weights, we selected the top 3% (amounting to n = 120) and computed the synchrony triggered averages for this bigger group as described in the main text. We find qualitatively the same results, see

Varying network size
In order to test sensitivity of the findings with respect to network size, apart from the network size of 5, 000 neurons that we used in the manuscript, we additionally simulated networks with 10, 000 (see Fig. K) and 20, 000 (see Fig. L

Topographic networks
Networks with local, distant-depended connectivity profiles are often taken as models for cortical connectivity [5]. To verify whether our findings hold for such types of networks, we simulated networks consisting of 10, 000 cells on the torus as described in the main text.
Here, we show some of their properties for the case of additive STDP rules. The results for partly or fully multiplicative STDP rules at excitatory synapses qualitatively look like the ones of the corresponding random networks.
The weights and rates distributions are very similar to the ones observed in the random network: rates are distributed approximately log-normal, the largest part of the weights distribution follows a power-low, see

Different learning rules
In order to assess the generality of the model, we exchange the plasticity rules and see if we obtain results (dynamical state, weight distributions, clustering of strong weights and emergence of driver neurons) qualitatively similar to the base model.

Partly multiplicative STDP
Exchanging the additive STDP rule at E-E synapses with a partly multiplicative one characterized by additive potentiation and multiplicative depression [2] ("van Rossum STDP", see Section 3) results in a network that still expresses long tailed distributions of firing rates and longtailed distributions of the excitatory weights, albeit this distribution being much less widespread than in the base model, Fig. N. As in the previous studies [2], we observe that synaptic weights distribution becomes unimodal, and the tail is no longer power-law distributed, but log-normal. Here, the largest EPSP sizes are much smaller than the largest ones in base model. This is due to the fact that in contrast to additive STDP, stronger synapses are subject to much stronger LTD under this rule [2].
Yet we still observe a clustering of the strongest outgoing synapses on cells with higher firing rates (see Fig. O, right), leading to the emergence of driver cells with a high mean outgoing weight. So although in this case the dynamical effect of driver neurons on their postsynaptic networks is much less pronounced due to the smaller absolute synaptic weights compared to the base model, they still exist (see Fig. P).

Multiplicative STDP
Exchanging the additive STDP rule at E-E synapses with a fully multiplicative one (see Section 3) results in a network expressing similar rate and weight distributions as in the case of a van Rossum STDP rule described previously, see Section 12.1.
The weight distribution again is unimodal with a log-normal tail, see   strongest outgoing synapses on cells with higher firing rates (see Fig. R, right), leading to the emergence of driver cells with a high mean outgoing weight.
As in the case for van Rossum STDP, the dynamical effect of driver neurons on their postsynaptic networks is much less pronounced compared to the case of additive STDP, but it is still present.

Synaptic scaling
A classical form of homeostatic plasticity called synaptic scaling acts on timescales of hours to days [6]. In the base model, homeostatic plasticity is implemented as synaptic weight normalization at the postsynaptic site of E-E connections. Instead of this homeostatic rule acting at fast time scales we also considered a synaptic scaling acting of the form acting on much slower time scales that was analyzed in [7], where w denotes the weight of a synapse, ν denotes the firing rate of the postsynaptic cell (taken as a running average over some time window having a length of a few seconds or minutes), ν 0 is a target firing rate and η a learning rate.
For the simulations we performed the updates according to Eq. 5 at fixed time intervals of 50 ms and chose a learning rate γ = 10 −6 along with ν 0 = 0 Hz. Firing rates were computed using a sliding window of length 100 s. Note that the target rate is never attained by any of the cells due to external and recurrent input. This setup yields similar weight distributions and clustering characteristics of strong synapses that are comparable to the base model (see Fig. T We observe that in this altered model convergence of synaptic weights takes much longer than in the base model (convergence took circa 15, 000 s in the example).

Clustering of strong synapses
In order to more closely examine the clustering of strong outgoing synapses at single cells, we define a class of strong synapses as the excitatory synapses having efficacies larger than µ + 3σ where µ and σ denote the mean and standard deviation of all excitatory synapses, see Tab. 1. We subsequently calculate the fraction of strong synapses at each cell and show histograms of these distributions for various configurations in Fig. U. Note that the tail in the network without inhibitory STDP is much smaller than in the case of a network including inhibitory STDP. The clustering effect is even more pronounced for the cases of van Rossum and multiplicative STDP rules.