Hebbian changes of excitatory synapses are driven by and further enhance correlations between pre- and postsynaptic activities. Hence, Hebbian plasticity forms a positive feedback loop that can lead to instability in simulated neural networks. To keep activity at healthy, low levels, plasticity must therefore incorporate homeostatic control mechanisms. We find in numerical simulations of recurrent networks with a realistic triplet-based spike-timing-dependent plasticity rule (triplet STDP) that homeostasis has to detect rate changes on a timescale of seconds to minutes to keep the activity stable. We confirm this result in a generic mean-field formulation of network activity and homeostatic plasticity. Our results strongly suggest the existence of a homeostatic regulatory mechanism that reacts to firing rate changes on the order of seconds to minutes.

Learning and memory in the brain are thought to be mediated through Hebbian plasticity. When a group of neurons is repetitively active together, their connections get strengthened. This can cause co-activation even in the absence of the stimulus that triggered the change. To avoid run-away behavior it is important to prevent neurons from forming excessively strong connections. This is achieved by regulatory homeostatic mechanisms that constrain the overall activity. Here we study the stability of background activity in a recurrent network model with a plausible Hebbian learning rule and homeostasis. We find that the activity in our model is unstable unless homeostasis reacts to rate changes on a timescale of minutes or faster. Since this timescale is incompatible with most known forms of homeostasis, this implies the existence of a previously unknown, rapid homeostatic regulatory mechanism capable of either gating the rate of plasticity, or affecting synaptic efficacies otherwise on a short timescale.

Funding: FZ was supported by the European Community's Seventh Framework Program under grant agreement no. 237955 (FACETS-ITN) and 269921 (BrainScales). GH was supported by the Swiss National Science Foundation. WG acknowledges funding from the European Research Council (no. 268689). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2013 Zenke et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

In summary we show that the stability of the background state requires the ratio between the timescales of homeostasis and plasticity to be smaller than a critical value which is determined by the network properties. For realistic network and plasticity parameters this requires the homeostatic timescale to be short, meaning that homeostasis has to react quickly to changes in the neuronal firing rate (on the order of seconds to minutes). Our results suggest that plasticity must either be gated rapidly by a third factor, or be accompanied by a yet unknown homeostatic control mechanism that reacts on a short timescale.

Here we formalize the link between stability of network activity and the timescales involved in homeostasis in the presence of Hebbian plasticity. We first study the stability of the background state during long episodes of ongoing plasticity in direct numerical simulations of large balanced networks with a metaplastic triplet STDP rule [30] in which the timescale of homeostasis is equal to the one of the rate detector. This allows us to determine the critical timescale beyond which stability is lost. In a second step we reduce the system to a generic two-dimensional mean-field model amenable to analytical considerations. Both the numerical and the analytical approach show that homeostasis has to react to rate changes on a timescale of seconds to minutes. We then show analytically and in simulations that these stability requirements are not specific to metaplastic triplet STDP, but generalize to the case of triplet STDP in conjunction with synaptic scaling.

To stabilize neuronal activity, homeostatic control mechanisms have been proposed theoretically [13] – [19] and various forms have indeed been found experimentally [20] – [22] . The term homeostasis comprises any compensatory mechanism that stabilizes neural firing rates in the face of plasticity induced changes. This includes compensatory changes in the overall synaptic drive (e.g. synaptic scaling [21] ), the neuronal excitability (intrinsic plasticity [23] ) or changes to the plasticity rules themselves (i.e. metaplasticity [20] ). Common to all experimentally found homeostatic mechanisms is their relatively slow response compared to plasticity. While synaptic weights can change on the timescale of seconds to minutes [24] – [26] , noticeable changes caused by homeostasis generally take hours or even days [21] , [27] – [29] . This is thought to be crucial since it allows neurons to detect their average firing rate by integrating over long times. While fluctuations on short timescales cause Hebbian learning and alter synapses in a specific way to store information, at longer timescales homeostasis causes non-specific changes to maintain stability [23] . The required homeostatic rate detector acts as a low-pass filter and therefore induces a time lag between the rate estimate and the true value of neuronal activity. As a result, homeostatic responses based on this detector become inert to sudden changes. The longer the filter time constant is, the more sluggish the homeostatic response becomes.

The awake cortex is constantly active, even in the absence of external inputs. This baseline activity, commonly referred to as the “background state”, is characterized by low synchrony at the population level and highly irregular firing of single neurons. While the direct implications of the background state are presently unknown, several neurological disorders such as Parkinson's disease, epilepsy or schizophrenia have been linked to various disruptions thereof [1] – [5] . Theoretically, the background state is currently understood as the asynchronous and irregular (AI) firing regime resulting from a dynamic balance of excitation and inhibition in recurrent neural networks [6] – [9] . Balanced networks exhibit low activity and small mean pairwise correlations [7] , [9] . However, even small changes in the amount of excitation can disrupt the background state [7] , [10] . Changes in excitation can arise from Hebbian plasticity of excitatory synapses: Subsets of jointly active neurons form strong connections with each other which is thought to be the neural substrate of memory [11] . However, Hebbian plasticity has the unwanted side effect of further increasing the excitatory synaptic drive into cells that are already active. The emergent positive feedback loop renders this form of plasticity unstable and makes it hard to reconcile with the stability of the background state [12] .

Results

In the following we first discuss our results obtained from simulating spiking neural networks in the balanced state with a Hebbian learning rule subject to a plausible learning rate. In the beginning we focus on a metaplastic mechanism that regulates the amount of synaptic long term depression (LTD) homeostatically. By systematically varying the time constant of the homeostatic rate detector, we find that stability of the background state requires homeostasis to act on a timescale of minutes. We then strive to understand the underlying mechanism of the instability from a generic mean field model, which we use to analytically confirm the critical time constant found in the spiking network simulations. Finally, to explore the generality of this mean field approach, we apply the analysis to two variations of the triplet learning rule. First, we add a slow weight decay to metaplastic triplet STDP and second we switch from homeostatic metaplasticity to synaptic scaling in combination with triplet STDP. In both cases we confirm analytically and in simulations that a fast rate detector is required to assure stability.