Abstract
The critical state is assumed to be optimal for any computation in recurrent neural networks, because criticality maximizes a number of abstract computational properties. We challenge this assumption by evaluating the performance of a spiking recurrent neural network on a set of tasks of varying complexity at  and away from critical network dynamics. To that end, we developed a plastic spiking network on a neuromorphic chip. We show that the distance to criticality can be easily adapted by changing the input strength, and then demonstrate a clear relation between criticality, taskperformance and informationtheoretic fingerprint. Whereas the informationtheoretic measures all show that network capacity is maximal at criticality, only the complex tasks profit from criticality, whereas simple tasks suffer. Thereby, we challenge the general assumption that criticality would be beneficial for any task, and provide instead an understanding of how the collective network state should be tuned to task requirement.
Introduction
A central challenge in the design of an artificial network is to initialize it such that it quickly reaches optimal performance for a given task. For recurrent networks, the concept of criticality presents such a guiding design principle^{1,2,3,4,5,6,7}. At a critical point, typically realized as a secondorder phase transition between order and chaos or stability and instability, a number of basic processing properties are maximized, including sensitivity, dynamic range, correlation length, information transfer, and susceptibility^{8,9,10,11,12}. Because all these basic properties are maximized, it is widely believed that criticality is optimal for task performance^{1,2,4,5,6,7,9}.
Tuning a system precisely to a critical point can be challenging. Thus ideally, the system selforganizes to criticality autonomously via locallearning rules. This is indeed feasible in various manners by modifying the synaptic strength depending on the pre and postsynaptic neurons’ activity only^{6,9,13,14,15,16,17,18,19}. The locality of the learning rules is key for biological and artificial networks where global information (e.g., taskperformance error or activity of distant neurons) may be unavailable or costly to distribute. Recently, it has been shown that specific locallearning rules can even be harnessed more flexibly: a theoretical study suggests that recurrent networks with local, homeostatic learning rules can be tuned toward and away from criticality by simply adjusting the input strength^{17}. This would enable one to sweep the entire range of collective dynamics from subcritical to critical to bursty, and assess the respective task performance.
Complementary to tuning collective network properties such as the distance to criticality, locallearning also enables networks to learn specific patterns or sequences^{20,21}. For example, spiketiming dependent plasticity (STDP) shapes the connectivity, depending only on the timing of the activity of the pre and postsynaptic neuron. STDP is central for any sequence learning—a central ingredient in language and motor learning^{20,21}. Such learning could strongly speed up convergence, and enables a preshaping of the artificial network—akin to the shaping of biological networks during development by spontaneous activity.
Given diverse learning rules and task requirements, it may be questioned whether criticality is always optimal for processing, or whether each task may profit from a different state, as hypothesized in ref. ^{10}. One could speculate that, e.g., the long correlation time at criticality on the one hand enables long memory retrieval, but on the other hand could be unfavorable if a task requires only little memory. However, the precise relation between the collective state, and specific task requirements is unknown.
When testing networks, the observed network performance is expected to depend crucially on the choice of the task. How can one then characterize performance independently of a specific task, such as classification or sequence memory? A natural framework to characterize and quantify processing of any local circuit in a taskindependent manner builds on information theory^{22,23}: classical information theory enables us to quantify the transfer of information between neurons, the information about the past input, as well as the storage of information. The storage of information can be measured within the network or as read out from one neuron. In addition, most recently classical mutual information is being generalized to more than two variables within the framework of partial information decomposition (PID)^{23,24,25,26}. PID quantifies the unique and redundant contribution of each source variable to a target, but most importantly also enables a rigorous quantification of synergistic computation, a key contributor for any information integration^{23,24,25,27,28}. Thereby information theory is a key stepping stone when linking local computation within a network, with global task performance.
Simulations of recurrent networks with plasticity become very slow with increasing size, because every membrane voltage and every synaptic strength has to be updated. Here, neuromorphic chips promise an accelerated and scalable alternative to neural network simulations^{29,30,31,32,33,34,35}. To achieve an efficient implementation, physical emulation of synapses and neurons in electrical circuitry are very promising^{36}. In such “neuromorphic chips,” all neurons operate in parallel, and thus the speed of computation is largely independent of the system size, and is instead determined by the time constants of the underlying physical neuron and synapse models—such as in the brain. Realizing such an implementation technically remains challenging, especially when using spiking neurons and flexible synaptic plasticity. The BrainScaleS 2 prototype system combines physical models of neurons and synapses^{37} with a general purpose processor carrying out plasticity^{38}. In this system, the analog elements provide a speedup, energy efficiency, and enable scaling to very large systems, whereas the general purpose processor enables to set the desired learning rules flexibly. Thus with this neuromorphic chip, we can run the longterm learning experiments—required to study the network selforganization—within very short computetime.
In the following, we investigate the relation between criticality, taskperformance and informationtheoretic fingerprint. To that end, we show that a spiking neuromorphic network with synaptic plasticity can be tuned toward and away from criticality by adjusting the input strength. We show that criticality is beneficial for solving complex tasks, but not the simple ones—challenging the common notion that criticality in general is optimal for computation. Methods from classical information theory as well as the novel framework of PID show that our networks indeed enfold their maximum capacity in the vicinity of the critical point. Moreover, the lagged mutual information between the stimulus and the activity of neurons allows to establish a relation between criticality (as set by the input strength) and task performance. Thereby, we provide an understanding how basic computational properties shape task performance.
Results
Model overview
We emulate networks of leaky integrateandfire (LIF) neurons on the mixedsignal neuromorphic prototype system BrainScaleS 2, which has N = 32 neurons (Fig. 1 a, b). Future versions of the BrainScaleS 2 chip will feature 512 neuron circuits with adaptiveexponential LIF dynamics and intercompartmental conductances. We use the term emulation in order to clearly distinguish between the physical implementation, where each observable has a measurable counterpart on the neuromorphic chip, and standard software simulations on conventional hardware. The system features an array of 32 × 32 currentbased synapses, where 20% of the synapses are programmed to be inhibitory. Synaptic plasticity acts equally on all synapses and is composed of a positive drift and a negative anticausal STDP term. In conjunction both terms lead to homeostatic regulation and thus stable network activity of about 20 Hz per neuron (Supplementary Fig. 1a). Plasticity is executed by an onchip general purpose processor alongside to the analog emulation of neurons and synapses. This allows for an uninterrupted and fast data acquisition. Even for the small prototype system, the advantages of neuromorphic computing in terms of speed and energy efficiency become important as depicted in ref. ^{39}.
Neurons are potentially alltoall connected, but K_{ext} out of the N synapses per neuron are used to inject external Poisson or pattern input. Effectively, K_{ext} quantifies the input strength with the extreme cases of K_{ext}/N = 1 for a feedforward network and K_{ext}/N = 0 for a fully connected recurrent network, which is completely decoupled from the input. Depending on the degree of external input K_{ext}, the network shows diverse dynamics (Fig. 1c, d). As expected^{17}, K_{ext} shapes the collective dynamics of the network from synchronized for low K_{ext} to more asynchronous irregular for high K_{ext}.
Critical dynamics arise under low input K _{ext}
The transition to burstiness for low K_{ext} suggests the emergence of critical dynamics, i.e., dynamics expected at a nonequilibrium secondorder phase transition. Indeed, as detailed in the following, we find signatures of criticality in the classical avalanche distributions (Figs. 2 and 3) as well as in the branching ratio (Fig. 4a), the autocorrelation time (Fig. 4b), the susceptibility, and in trialtotrial variations (Fig. 4d).
To test whether the network indeed approaches criticality, we assume the established framework of a branching process^{8,40,41,42}. In branching processes, a spike at time t triggers on average m postsynaptic spikes at time t + 1, where m is called the branching parameter. For m = 1 the process is critical, and the dynamics give rise to large cascades of activity, called avalanches^{8,43}. The size s of an avalanche is the total number of spikes in a cluster and is power law distributed at criticality. The binwidth for the estimation of the underlying distributions is set to the mean interevent interval following common methods^{44}. Our network shows power law distributed avalanche sizes s over two orders of magnitude for low K_{ext} (Fig. 2a). For almost any K_{ext}, the distribution is better fitted by a power law than by an exponential distribution^{45} (Fig. 2d). However, only for low K_{ext} the exponent of the avalanche distribution is close to the expected one, α_{s} ≈ 1.5 (Fig. 2c), and the power law shows the largest cutoff s_{cut} (Fig. 2b). For low K_{ext}, the networks tend to get unstable due to the limited number of neurons explaining the decline in s_{cut} (Fig. 2b) and in the maximumlikelihood comparison (Fig. 2d). Together, all the quantitative assessment of the avalanches indicate that a low degree of input K_{ext} produces criticallike behavior.
In a control experiment, we investigate finitesize scaling in software simulations, as the current physical system features only 32 neurons. Therefore, a network with the same topology, plasticity rules, and singleneuron dynamics (though without parameter noise and hardware constraints) is simulated for various system sizes N. The resulting avalanche distributions show power laws for any system size (Fig. 3a), and the cutoff s_{cut} scales with N as expected at criticality (Fig. 3b). The scaling exponent is 1.6 ± 0.2. Together, these numerical results confirm the hypothesis that for low degrees of input K_{ext}, the small network that is emulated on the chip selforganizes as close to a critical state as possible.
The implementation on neuromorphic hardware promises fast emulation. Already for N = 32, the neuromorphic chip is about a factor of 100 faster than the Brian 2 simulation. To give numbers, a single plasticity experiment with a duration of 600 s biological time is simulated in 570 s on a single core of a Intel Xeon E52670 CPU in Brian 2, but emulated in only 6 s on the neuromorphic chip. Hence, a neuromorphic implementation is very promising especially for the future full size chip: When running such detailed networks as classical simulations, the computational overhead scales with \({\mathcal{O}}({N}^{2})\) due to the alltoall connectivity and synaptic plasticity on conventional hardware. In contrast, for the neuromorphic system, the execution time is largely independent of the system size N, as long as the network can be implemented on the system.
The assumption that the critical state of the network corresponds to the universality class of critical branching processes is tested further by properly inferring the branching parameter m (Eq. (17)), the autocorrelation times, and the response to perturbations. First, the branching parameter m characterizes the spread of activity and is smaller (larger) than unity for subcritical (supercritical) processes. For our model, it is always in the subcritical regime, but tends toward unity for low K_{ext} (Fig. 4a). Second, the autocorrelation time τ_{corr} is expected to diverge at criticality as \({\tau }_{{\rm{branch}}}(m) \sim {\mathrm{lim}\,}_{m\to 1}(1/\mathrm{log}\,(m))=\infty\)^{42}. Indeed, τ_{corr} as estimated directly from the autocorrelation of the population activity is maximal for low K_{ext} (Fig. 4b). Third, the estimates of m and τ_{corr} are in theory related via the analytical relation \({\tau }_{{\rm{branch}}}(m) \sim 1/\mathrm{log}\,(m)\). This relation holds very precisely in the model (Fig. 4c, correlation coefficient ρ = 0.998, p < 10^{−10}). Fourth, toward criticality, the response to any perturbation increases. The impact of a small perturbation is quantified by a variant of the vanRossum distance Δ_{VRD} (Eq. (15)). It peaks for low degrees of the external input K_{ext} (Fig. 4d). Last, one advantage of operating in the vicinity of a critical point is the ability to enhance stimulus differences by the system response. This is reflected in a divergence of the susceptibility at the critical point. The susceptibility χ (Eq. (16)), quantified here as the change in the population firing rate in response to a burst of N_{pert} = 6 additional spikes, is highest for low K_{ext} (Fig. 4d). Thus overall, the avalanche distributions as well as the dynamic properties of the network all indicate that it selforganizes to a critical point under low degree of input K_{ext}.
Network properties have to be tuned to task requirements
It is widely assumed that criticality optimizes task performance. However, we found that one has to phrase this statement more carefully. While certain abstract computational properties, such as the susceptibility, sensitivity, or memory time span are indeed maximal or even approaching infinity at a critical state, this is not necessary for task performance in general^{5,11,42,46}. We find that it can even be detrimental. For every single task complexity, a different distance to criticality is optimal, as outlined in the following.
We study the performance of our recurrent neural network in the framework of reservoir computing: the performance of a recurrent neural network is quantified by the ability of linear readout neurons to separate different sequences^{47,48,49}. To that end, it is often necessary to maintain information about past input for long time spans. To test performance, we specifically use a nbit sum and a nbit parity task and trained a readout on the activity of N_{read} = 16 randomly chosen neurons of the reservoir. For the two given tasks complexity increases with n: to solve the tasks, the network has to both memorize and process the input from the n past steps. As reservoirs close to a critical point have longer memory as quantified by the lagged mutual information (I_{τ}, Fig. 7), one expects that particularly the memory intensive tasks profit from criticality (tasks with high n are better at low degrees of input K_{ext}). In contrast, simple tasks (low n) might suffer from criticality because of the maintenance of memory about unnecessary input. Since the estimation of parity, in contrast to the sum, is fully nonlinear, their direct comparison allows to further dissect task complexity. Thus, depending on the task complexity, there should be an ideal K_{ext}, leading to maximal performance.
For our network, we find indeed that maximal task performance depends on both, task complexity and distance to criticality: simple sum tasks (n = 5) are optimally solved away from criticality, whereas complex sum tasks (n = 25) profit from the long timescales arising at criticality (Fig. 5a). The nonlinear parity task profits even more from criticality: even for n = 5 networks closer to the critical point promote task performance (Fig. 5b). Hence, we are capable of adapting the networks computational properties to task complexity by finetuning the strength of the input.
Likewise, we investigated the ability of our networks to do combinations of operations by considering the classic nonlinear autoregressive moving average (NARMA) task^{48}. The network’s peak performance again moves toward criticality with higher task complexity (Fig. 5e).
To further tune the difficulty of task, we reduced the number of neurons visible to the readout N_{read}. We expect that in principle information about, e.g., parity could be available in a single neuron if the network is sufficiently close to criticality, because critical network dynamics are not only characterized by temporal, but also spatial correlations. The ability to condense information about extended stimuli in the activity of few neurons can be valuable. To quantify the effect of spatial correlations on computation, we trained linear classifiers on the activity of a subset of neurons for the 5bit sum and the parity task. When lowering N_{read} from 8 to 4, only the nonlinear parity tasks increasingly profits from critical network dynamics (Fig. 5c, d). In contrast, the information necessary to solve the linear sum task seems to be globally available in the network response even for subcritical dynamics. The ability to locally read out global information from the network is of equal importance for both, large neuromorphic systems^{29} and living networks^{50,51}.
Adaptation to task by dynamic switching of input strengths
We know from the previous experiments that for high n, the nbit parity task is solved best at criticality, whereas for low n, the subcritical regime leads to best performance. In the following, we investigate how to transit between both states. To achieve this, we take the state of a critical network and switch the degree of the input K_{ext} to a subcritical configuration and vice versa. The performance is evaluated after various numbers of synaptic updates. This task switch generates the same working points as the previous emulations that start with synaptic weights w_{ij} = 0 ∀ i, j and have a long adaptation phase (red stars in Fig. 6).
A fast adaption to different input strengths is required to switch between tasks of different complexity. The transition from critical to subcritical is achieved after the application of about 50 synaptic updates corresponding to 50 s biological time, whereas going from subcritical to critical takes about 500 updates and therefore 500 s (Fig. 6). However, due to the speedup of the neuromorphic chip, the adaptation takes only about 0.5 s wall clock time and can even be lowered by decreasing the integration time over spike pairs in the synaptic update rule. As alternative strategies, one could switch between saved configurations, or run a hierarchy of networks with different working points in parallel^{52}.
Taskindependent quantification of computational properties
While task performance is the standard benchmark for any model, such benchmark tasks have two disadvantages: in many biological systems, such as higher brain areas or in vitro preparations, such tasks cannot be applied. Even if tasks can be applied, the outcome will always depend on the chosen task. To quantify computational properties in a taskindependent manner, information theory offers powerful tools^{23}. Using the Poisson noise input, we find that the lagged mutual information I_{τ} between the input s_{i} and the activity of a neuron after a time lag τ, a_{j} predicts the performance on the parity task. Here, at high K_{ext} (away from criticality) information about the input is maximal for very short τ, but decays very quickly (Fig. 7a). This fast forgetting is important to irradiate past, taskirrelevant input that would interfere with novel, taskrelevant input. At small K_{ext}, the recurrence is stronger and input can be read out for much longer delays (20 ms vs. 60 ms). This active storage of information is required in a reservoir to solve any task that combines past and present input, and hence the more complex parity task also profits from it. However, the representation of input in every single neuron becomes less reliable (i.e., I_{τ} is smaller). A measure for the representation of the input in the network could be obtained by integrating I_{τ} over τ. Interestingly, this memory capacity (MC) stays fairly constant (Fig. 7b). Note that we only quantified the representation of the input in a single neuron, a measure very easily accessible in experiments; obviously the readout can draw on the distributed memory across all neurons, which jointly provide a much better readout.
The memory maintenance for task processing has to be realized mainly by activity propagating on the recurrent connections in the network. Therefore, it is often termed active information storage (AIS)^{53}. The recurrent connections become stronger closer to criticality, and as a consequence we find that the lagged mutual information between pairs of neurons in the reservoir also increases (Fig. 7c). As a result the MC of the reservoir increases over almost two orders of magnitude when approximating criticality (lower K_{ext}, Fig. 7d). This increase in internal MC carries the performance on the more complex parity tasks.
When assessing computational capacities, information theory enables us to quantify not only the entropy (H) and mutual information (I) between units, but also to disentangle transfer and storage of information, as well as unique, redundant, and synergistic contributions of different source neurons^{23,24,25,27,54}. We find that all these quantities increase with approaching criticality (smaller K_{ext}, Figs. 8b and 9c). This indicates that the overall computational capacity of the model increases, as predicted for the vicinity of the critical state^{1,2,7,11}.
In more detail, the AIS of a neuron, as well as the I and the transfer entropy (TE) between pairs of neurons increase with lower K_{ext} (Fig. 8b). In our analysis, these increases reflect memory that is realized as activity propagation on the network, and not storage within a single neuron, because the binsize used for analysis is larger than the refractory period τ_{ref}, synaptic τ_{syn}, and membranetimescales τ_{m}. Information theory here enables us to show that active transfer and storage of information within the network strongly increases toward criticality. A similar increase in I, AIS, and TE has been observed for the Ising model and reservoirs at criticality^{1,11}, and hence supports the notion that criticality maximizes information processing capacity. Note however, that this maximal capacity is typically not necessary; as shown here, it can even be unfavorable when solving simple tasks.
Very recently, it has become possible to dissect further the contributions of different neurons to processing, using PID^{24}. PID enables us to disentangle for a target neuron a_{i}, how much unique information it obtains from its own past activity \({{\bf{a}}}_{{\bf{i}}}^{}\), or the past activity of a second neuron \({{\bf{a}}}_{{\bf{j}}}^{}\); and how much information is redundant or even synergistic from the two (Fig. 9a). Synergistic information is that part of information that can only be computed if both input variables are known, whereas redundant information can be obtained from one or the other.
All the PID components increase when approaching the critical point (low K_{ext}, Fig. 9c). Quantitatively, the redundant and the synergistic information are always stronger than the unique ones that are about ten times less. The shared information dominates closer to criticality, mirroring the increased network synchrony and redundancy between neurons. Further, the synergistic contribution, i.e., the contributions that rely on the past of both neurons slightly increases, and is indeed the largest contribution for high K_{ext}. This reflects that typically the joint activity of both neurons is required to activate a LIF neuron. Interestingly, the strong increase in shared information (i.e., redundancy) does not seem to impede the performance at criticality (small K_{ext}). However, for even higher synchrony, as expected beyond this critical transition, the shared information might increase too much and thereby decrease performance.
Discussion
In this study, we used a neuromorphic chip to emulate a network, subject to plasticity, and showed a clear relation between criticality, taskperformance and informationtheoretic fingerprint. Most interestingly, simple tasks do not profit from criticality while complex ones do, showing that every task requires its own network state.
The state and hence computational properties can readily be tuned by changing the input strength, and thus a critical state can be reached without any parameter finetuning within the network. This robust mechanism to adapt a network to task requirements is highly promising, especially for large networks where many parameters have to be tuned and in analog neuromorphic devices that are subject to noise in parameters and dynamics.
It has been generally suggested that criticality optimizes task performance^{1,9}. We show that this statement has to be specified: indeed, criticality maximizes a number of properties, such as the autocorrelation time (Fig. 4b), the susceptibility (Fig. 4d), as well as informationtheoretic measures (Figs. 7–9). However, this maximization is apparently not at all necessary, potentially even detrimental, when dealing with simple tasks. For our simple task, high network capacity results in maintenance of taskirrelevant information, and thereby harms performance. This is underlined by our results that clearly show that all abstract computational properties are maximized at criticality, but only the complex tasks profit from criticality. Hence, every task needs its own state and therefore a specific distance to critical dynamics.
The input strength could not only be controlled by changing K_{ext}, the number of synapses of a neuron that were coupled to the input. An equally valid choice is a change of external input rate to each neuron. In fact, we showed that changing the input rate has the same effects on the relation between criticality (Supplementary Figs. 1–3), task performance (Supplementary Fig. 4), and information measures (Supplementary Figs. 5 and 6) as changing K_{ext}. Moreover, in this framework the lowest input rates even allow to cross the critical point (Supplementary Figs. 3 and 4). Thus for both control mechanisms or a combination, there exists an optimal input strength, where the homeostatic mechanisms bring the network closest to critical. This optimal input strength has been derived analytically for a meanfield network by Zierenberg et al.^{17}, and could potentially be used to predict the optimal input strength for other networks and tasks as well.
Not only the input strength, but also the strength of inhibition can act as a control parameter. Inhibition plays a role in shaping collective dynamics and is known to generate oscillations^{55,56}. For a specific ratio of excitation and inhibition, criticality has been observed in neural networks^{18,19,57,58}. Likewise, our networks has 20% inhibitory neurons. However, inhibition would not be necessary for criticality^{13,42}. Nevertheless, the existence of more than one control parameters (degree of input, input rate, and inhibition) allows for flexible adjustment even in cases where only one of them could be freely set without perturbing input coding.
Plasticity plays a central role in selforganization of network dynamics and computational properties. In our model, the plasticity, neuron and synapse dynamics feature quite some level of biological detail (Table 1), and thus results could potentially depend on them. All synaptic weights are determined by the synaptic plasticity. Here, we showed results for homeostasis and STDP that implement the negative (anticausal) arm only. When implementing the positive (causal) arm of STDP in addition, the network destabilized, despite counteracting homeostasis. This is a well known problem^{59}. Our implementation is still similar to full STDP, because anticausal correlations are weakened and the causal ones are indirectly strengthened by homeostasis. With its similarity to STDP and its inherent stability, our reduced implementation may be useful for future studies.
The characterization of the network in a taskdependent as well as in a taskindependent manner is essential for understanding the impact of criticality on computation. The computational properties in the vicinity of a critical point have been investigated by the classical measures AIS, I, and TE alone^{5,60}, or by PID alone^{27,61}. In this paper, we indeed showed that criticality maximizes capacity, but this does not necessarily translate to maximal task performance. Moreover, the lagged I between the stimulus and the activity of neurons allows to estimate memory timescales required to solve our tasks. This enables us to understand how task complexity and the informationtheoretic fingerprint are related. Such understanding is the basis for wellfounded design decisions of future artificial architectures.
The presented framework is particularly useful for analog neuromorphic devices as analog components have inherent parameter noise as well as thermal noise, which potentially destabilize the network. Here, the synaptic plasticity plays a key role in equalizing out particularly the parameter noise, as also demonstrated for shortterm plasticity^{62}, and thus makes knowledge about parameter variations, as well as specific calibration to some extend unnecessary.
Despite the small system size (N = 32 neurons only), the network not only showed signatures of criticality, but also developed quite complex computational capabilities, reflected in both, the task performance and the abstract informationtheoretic quantities. We expect that a scaleup of the system size would open even richer possibilities. Such a scaleup would not even require finetuning of parameters, as the network selftunes owing to the locallearning rules. As soon as larger chips are available, we expect that the abilities of neuromorphic hardware could be exhausted in terms of speed and energy efficiency allowing for long, largescale, and powerful emulations.
Overall, we found a clear relation between criticality, taskperformance and informationtheoretic fingerprint. Our result contradicts the widespread statement that criticality is optimal for information processing in general: while the distance to criticality clearly impacts performance on the reservoir task, we showed that only the complex tasks profit from criticality; for simple ones, criticality is detrimental. Mechanistically, the optimal working point for each task can be set very easily under homeostasis by adapting the mean input strength. This shows how critical phenomena can be harnessed in the design and optimization of artificial networks, and may explain why biological neural networks operate not necessarily at criticality, but in the dynamically rich vicinity of a critical point, where they can tune their computational properties to task requirements^{10,63}.
Methods
We start with a description of the implemented network model, followed by a summary of the analysis techniques. All parameters are listed in Table 1 and all variables in the Supplementary Tables 1 and 2.
Model
The results shown in this article are acquired on the mixedsignal neuromorphic hardware system described in ref. ^{38} (Fig. 1b). In the following a brief overview of the model, which is approximated by the physical implementation on the hardware, and the programmed plasticity rule is given. Since the neuromorphic hardware system comprises analog electric circuits, transistor mismatch causes parameter fluctuations, which can be compensated by calibration^{37}. Here, no explicit calibration on the basis of single neurons and synapses is applied. Instead, only parameters common to all neurons/synapses are set such that all parts behave according to the listed equations, especially that all parts are sensible to input but silent in the absence of input. This choice leads to uncertainties in the model parameters as reported in Table 1.
Neurons: Implemented in analog circuitry, the neurons approximate currentbased LIF neurons. The membrane potential u_{j} of the jth neuron obeys
with the membrane time constant τ_{mem}, the leak conductance g_{leak} = C_{m}/τ_{mem}, the leak potential u_{leak,} and the input current I_{j}(t). The kth firing time of neuron j, \({t}_{j}^{k}\), is defined by a threshold criterion
Immediately after \({t}_{j}^{k}\), the membrane potential is clamped to the reset potential u_{j}(t) = u_{reset} for \(t\in \left({t}_{j}^{k},{t}_{j}^{k}+{\tau }_{{\rm{ref}}}\right]\), with the refractory period τ_{ref}. The neuromorphic hardware system comprises N = 32 neurons, operating in continuous time due to the analog implementation.
Synapses: Like the membrane dynamics, the synapses are implemented in electrical circuits. Each neuron features N = 32 presynaptic partners (indegree is 32). The synaptic input currents onto the jth neuron enter the neuronal dynamics in Eq. (1) as the sum of the input currents of all presynaptic partners i, \({I}_{j}(t)=\mathop{\sum }\nolimits_{i = 1}^{N}{I}_{ij}(t)\), where I_{ij}(t) is given by
with the excitatory and the inhibitory synaptic time constants \({\tau }_{{\rm{syn}}}^{{\rm{exc}}}\) and \({\tau }_{{\rm{syn}}}^{{\rm{inh}}}\). N_{inh} synapses of every neuron j are randomly selected to be inhibitory. The external synaptic current \({I}_{ij}^{{\rm{ext}}}(t)\) depends on the lth spike time of an external stimulus i, \({s}_{i}^{l}\), whereas the recurrent synaptic current \({I}_{ij}^{{\rm{rec}}}(t)\) depends on the kth spike time of neuron i, \({t}_{i}^{k}\), each of which transmitted to neuron j
with the synaptic delay d_{syn} and the weight conversion factor γ. The synaptic weight from an external spike source i to neuron j is denoted by \({w}_{ij}^{{\rm{ext}}}\), and \({w}_{ij}^{{\rm{rec}}}\) is the synaptic weight from neuron i to neuron j. Every synapse either transmits external events \({s}_{i}^{l}\) or recurrent spikes \({t}_{i}^{k}\), i.e., if \({w}_{ij}^{{\rm{stim}}}\ge 0\) then \({w}_{ij}^{{\rm{rec}}}=0\) and vice versa.
Network: The LIF neurons are potentially connected in an alltoall fashion. A randomly selected set of K_{ext} synapses of every neuron is chosen to be connected to the spike sources. As every synapse could either transmit recurrent or input spikes, the K_{ext} synapses do not transmit recurrent spikes.
Plasticity: In the network, all synapses are plastic, the recurrent and the ones linked to the external input. Therefore, we skip the superscript of the synaptic weight and drop the distinction of \({t}_{i}^{k}\) and \({s}_{i}^{k}\) in the following description. Weights are subject to three contributions: a weight drift controlled by the parameter λ_{drift}, a correlation sensitive part controlled by λ_{stdp}, and positively biased noise contributions. This is very similar to STDP, however with specific depression, but unspecific potentiation. A specialized processor on the neuromorphic chip is programmed to update synaptic weights to w_{ij}(t + T) = w_{ij}(t) + Δw_{ij} according to
The STDPkernel function f depends on the pre and postsynaptic spike times in the time interval [t − T, t)
with \({t}_{i}^{k}\, > \, {t}_{j}^{l}\), and \({t}_{i}^{k},{t}_{j}^{l}\in [tT,t)\), and only nearestneighbor spike times are considered in the sum. η_{stdp} and τ_{stdp} denote the amplitude and the time constant of the STDPkernel. The term n_{ij}(t) adds a uniformly distributed, biased random variable
where n_{amp} specifies the range, while 〈n〉 is the bias of the random numbers.
The parameters λ_{stdp} and λ_{drift} are chosen such that the average combined force of the drift and the stochastic term is positive. Thus, only the negative arm of STDP is implemented.
Initialization: The synaptic weights are initialized to w_{ij} = 0 μA. Afterward, the network is stimulated by N Poissondistributed spike trains of rate ν by the K_{ext} synapses of every neuron. By applying Eq. (7) for the total duration T^{burnin} weights w_{ij} ≠ 0 μA develop. For every K_{ext}, the network is run 100 times, each with a different random seed. If not stated otherwise, the resulting weight matrices are used as initial conditions for experiments with frozen weights (Δw_{ij} = 0) for a duration of \({T}^{\exp }\) on which the analysis is performed.
Simulations: To complement the hardware emulations, an idealized version of the network is implemented in Brian 2^{64}. Specifically, no parameter or temporal noise is considered, and weights are not discretized as it is the case for the neuromorphic chip. For simplicity, the degree of the input is implemented probabilistically by connecting each neuroninput pair with probability K_{ext}/N and each pair of neurons with probability (N − K_{ext})/N.
Evaluation
Binning: The following measures rely on an estimate of activity, therefore we apply temporal binning
where δt corresponds to the binwidth, and 1 is the indicator function. With this definition, we are able to define the binarized activity for a single process i
The variable x_{i}(t) can represent either activity of a neuron in the network a_{i}(t), or of a stimulus spike train s_{i}(t), and correspondingly the spike times \({x}_{i}^{k}\) represent spikes of network neurons or stimulus (input) spike trains.
The population activity a(t) of the network is defined as
Neural avalanches: A neural avalanche is a cascade of spikes in neural networks. We extract avalanches from the population activity a(t), obtained by binning the spike data with δt corresponding to the mean interevent interval, following established definitions. In detail, one avalanche is separated from the subsequent one by at least one empty time bin^{43}. The size s of an avalanche is defined as the number of spikes in consecutive nonempty time bins. At criticality, the size distribution P(s) is expected to follow a power law.
To test for criticality, we compare whether a power law or an exponential distributions fits the acquired avalanche distribution P(s) better ^{45}. For the fitting, first the best matching distribution is determined based on the fitlikelihood. The fitrange is fixed to s ∈ {4, 3 ⋅ N} as the system is of finite size. An estimation of the critical exponent α_{s} and an exponential cutoff s_{cut} is obtained by fitting a truncated power law
for s ≥ 1. Power law fits are performed with the Python package power law described in ref. ^{65}.
Fano factor: The variability of the population activity is quantified by the Fano factor \(F={\sigma }_{a}^{2}/{\mu }_{a}\), where \({\sigma }_{a}^{2}\) is the variance and μ_{a} is the mean of the population activity a(t), binned with δt = τ_{ref}.
Trialtotrial variability and susceptibility: The trialtotrial distance Δ_{VRD} is obtained by stimulating the same network twice with the same Poisson spike trains, leading to two different trials m and n influenced by variations caused by the physical implementation. The resulting spike times in trial m emitted by neuron i, termed \({t}_{i,m}^{j}\), are convolved with a Gaussian
and likewise for trial n. The width is chosen to be σ_{VRD} = τ_{ref} and the temporal resolution for the integration is chosen to be 0.1 ms. From different trials m and n the distance is calculated
To obtain an estimate of the networks sensitivity χ to external perturbations, a pulse of N_{pert} additional spikes is embedded in the stimulating Poisson spike trains at time t_{pert}
normalized to the number of external connection \({K}_{{\rm{ext}}}^{2}\) to compensate for the decoupling from external input with decreasing K_{ext}. The population activity is estimated with binsize δt = d_{syn}. By evaluating χ immediately after the perturbation, only the effect of the perturbation is captured by minimizing the impact of trialtotrial variations.
To calculate χ and Δ_{VRD}, each weight matrix, obtained by the application of the plasticity rule, is used as initial condition for ten emulations with frozen weights and fixed seeds for the Poissondistributed spike trains of duration T^{pert} and T^{static}. In addition, a perturbation of size N_{pert} at t_{pert} = T^{pert}/2 is embedded for the estimation of χ.
Autoregressive model: Mathematically, the evolution of spiking neural networks is often approximated by a firstorder autoregressive representation. To assess the branching parameter m of the network in analogy to^{42,43}, we make use of the following ansatz:
where the population activity in the next time step, a(t + 1) is determined by internal propagation within the network (m), and by external input h. Here, 〈.∣.〉 denotes the conditional expectation and m corresponds to the branching ratio. For m = 1, the system is critical, for m > 1 the system is supercritical and activity grows exponentially on expectation (if not limited by finitesize effects), whereas for m < 1 the activity is stationary. The branching parameter m is linked to the autocorrelation time constant by \({\tau }_{{\rm{branch}}}=\delta t/\mathrm{ln}\,(m)\). To obtain the activity a(t) the binwidth δt is set to the refractory time τ_{ref}. Estimating m is straight forward here, as subsampling^{66,67} does not impact the estimate. Thus a classical estimator can be used, i.e., m is equal to the linear regression between a(t) and a(t + 1). For model validation purposes, the autocorrelation function ρ_{a,a} is calculated on the population activity a(t) binned with δt = τ_{ref}
where σ_{a} is the standard deviation, and μ_{a} the mean of the population activity. Subsequently, ρ_{a,a} is fitted by an exponential to yield the time constant τ_{corr}.
Information theory: We use notation, concepts and definitions as outlined in the review^{23}. In brief, the time series produced by two neurons represent two stationary random processes X_{1} and X_{2}, composed of random variables X_{1}(t) and X_{2}(t), t = 1, . . . , n, with realizations x_{1}(t) and x_{2}(t). The corresponding embedding vectors are given in bold font, e.g., \({{\bf{X}}}_{1}^{l}(t)=\{{X}_{1}(t),{X}_{1}(t1),...,{X}_{1}(tl+1)\}\). The embedding vector \({{\bf{X}}}_{1}^{l}(t)\) is constructed such that it renders the variable X_{1}(t + 1) conditionally independent of all random variables \({X}_{1}(t^{\prime} )\) with \(t^{\prime} <tl+1\), i.e., \(p({X}_{1}(t+1) {{\bf{X}}}_{1}^{l}(t),{X}_{1}(t^{\prime} ))=p({X}_{1}(t+1) {{\bf{X}}}_{1}^{l}(t))\). Here, (⋅∣⋅) denotes the conditional.
The entropy (H) and mutual information (I) are calculated for the random variables X_{1} and X_{2}, if not denoted otherwise. This is equivalent to using l = 1 above, e.g., H(X_{1}) and I(X_{1} : X_{2}) = H(X_{1}) − H(X_{1}∣X_{2}). We abbreviate the past state of spike train 1 by \({{\bf{X}}}_{1}^{}\): thus \({{\bf{X}}}_{1}^{l}(t1)=\{{X}_{1}(t1),X(t2),...,{X}_{1}(tl)\}\). The current value of the spike train is denoted by X_{1}. With this notation the AIS of, e.g., X_{1} is given by
In the same way, we define the TE between source X_{1} and target X_{2}
The lagged mutual information for time lag τ is defined as \({{\rm{I}}}_{\tau }({X}_{1}:{X}_{2})={{\rm{I}}}_{\tau }\left({X}_{1}(t):{X}_{2}(t+\tau )\right)\). Integrating the lagged I defines the MC
with a maximal delay N_{τ} = 100. The I of a sufficiently large N_{τ} is subtracted to account for potential estimation biases.
To access the information modification the novel concept of PID is applied^{24,25,27}. Intuitively, information modification in a pairwise consideration should correspond to the information about the present state of a process only available when considering both, the own process past and the past of a source process. Therefore, the joint mutual information \({\rm{I}}({X}_{1}:{{\bf{X}}}_{1}^{},{{\bf{X}}}_{2}^{})\) is decomposed by PID into the unique, shared (redundant), and synergistic contributions to the future spiking of one neuron, X_{1}, from its own past \({{\bf{X}}}_{1}^{}\), and the past of a second neuron or an input stimulus \({{\bf{X}}}_{2}^{}\): In more detail, we quantify

(1)
The unique information \({{\rm{I}}}_{{\rm{unq}}}({X}_{1}:{{\bf{X}}}_{1}^{}\setminus {{\bf{X}}}_{2}^{})\) that is contributed from the neurons own past.

(2)
The unique information \({{\rm{I}}}_{{\rm{unq}}}({X}_{1}:{{\bf{X}}}_{2}^{}\setminus {{\bf{X}}}_{1}^{})\) that is contributed from a different spike train (neuron or stimulus).

(3)
The shared information \({{\rm{I}}}_{{\rm{shd}}}({X}_{1}:{{\bf{X}}}_{2}^{};{{\bf{X}}}_{1}^{})\) that describes the redundant contribution.

(4)
The synergistic information \({{\rm{I}}}_{{\rm{syn}}}({X}_{1}:{{\bf{X}}}_{2}^{};{{\bf{X}}}_{1}^{})\), i.e., the information that can only be obtained when having knowledge about both past states.
I_{syn} is what we consider to be a suitable measure for information modification^{27}.
The joint mutual information as defined here is the sum of the AIS and the TE
We calculated H, AIS, I, and TE with the toolbox JIDT^{68}, whereas the PID was estimated with the BROJA2PID estimator^{69}. The activity is obtained by binning the spike data with δt = τ_{ref} and setting l to 4 to incorporate sufficient history. I and TE as well as the PID were calculated pairwise between all possible combinations of processes. Results are typically normalized by H to compensate for potential changes in the firing rate for changing values of K_{ext} (Supplementary Fig. 1). For the pairwise measures, H of the target neuron is used for normalization.
Reservoir computing: The performance of the neural network as a reservoir^{47,48} is quantified using a variant of the nbit parity task. The network weights are frozen (i.e., plasticity is disabled) to ensure that the network state is not changed by the input.
To solve the parity requires to classify from the network activity a_{j}(t), whether the last n bits of input carried an odd or even number of spikes. The network is stimulated with a single Poissondistributed spike train of frequency ν acting equally on all external synapses, i.e., the input spike times are \({s}_{i}^{k}={s}^{k}\ \forall \ i\). Spike times are binned according to Eq. (11) with binwidth δt to get a measure of the n past bits. The resulting stimulus activity s(t) is used to calculate the nbit parity function according to
with \({p}_{n}\left[s(t)\right]\in \{0,1\}\) and the modulus 2 addition ⊕ , i.e., whether an odd or even number of spikes occurred in the n past time steps of duration δt.
On the activity a_{j}(t) of a randomly selected subset \({\mathcal{U}}\) of neurons with cardinality N_{read} a classifier is trained
where Θ(⋅) is the Heaviside function, and v(t) is the predicted label. The weight vector w_{j} of the classifier is determined using linear regression on a set of training data s_{train} of duration T^{train}
The network’s performance on the parity task is quantified by \({\rm{I}}\left({p}_{n}\left[{s}_{{\rm{test}}}(t)\right],v(t)\right)\) on a test data set s_{test} of duration T^{test}. The performance I is offset corrected by training the very same classifier on a shuffled version of p_{n}[s(t)]. Moreover, we weighted each sample in the regression in Eq. (25) with the relative occurrence of their respective class to compensate for imbalance. Temporal binning with δt = 1 ms is applied to s_{train}, s_{test}, as well as a_{j}(t).
In a second task, the stimulus activity s(t) is used to calculate the nbit sum according to
i.e., how many spikes occurred in the n past time steps of duration δt. Here, the classifier described above is extended to multiple classes by adding readout units. The decision of the classifier is implemented by a winnertakeall mechanism across units.
In a third task, the readout is trained to calculate the NARMA system x_{n}(t)
with α = 0.3, β = 0.05, γ = 1.5, δ = 0.1^{70}, and the normalized stimulus activity
Again, a linear classifier is trained on the network activity
Here, the performance is quantified by the normalized rootmeansquare error (NRMSE)
with the standard deviation of the vote of the linear classifier σ_{y}.
Data availability
Data available on request from the authors.
Code availability
Code available on request from the authors.
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Acknowledgements
We thank M. Loidold, J. Pinheiro Neto, K. Schreiber, P. Spitzner, and J. Zierenberg for helpful comments on the manuscript. The authors express their gratitude toward S. Billaudelle, E. Müller, J. Weis, and Y. Stradmann for their support and helpful discussions. This work has received funding from the European Union Sixth Framework Programme ([FP6/20022006]) under grant agreement no 15879 (FACETS), the European Union Seventh Framework Programme ([FP7/20072013]) under grant agreement no. 604102 (HBP), 269921 (BrainScaleS), and 243914 (BrainiNets), and the Horizon 2020 Framework Programme ([H2020/20142020]) under grant agreement no 720270 and 785907 (HBP), as well as the Manfred Stärk Foundation. V.P. was supported by the Max Planck Society. M.W. has received funding from the VolkswagenStiftung. The authors acknowledge support by the state of BadenWürttemberg through bwHPC.
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B.C., D.S., and V.P. conceptualized the work. B.C. and V.P. wrote the manuscript. B.C. conducted the experiments. D.S. contributed to many of the experiments. M.K. contributed to the parity task. M.W. contributed to the information theory. V.P. supervised the work. J.S. is the architect and lead designer of the neuromorphic platform. J.S. and K.M. provided conceptual and scientific advice.
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Cramer, B., Stöckel, D., Kreft, M. et al. Control of criticality and computation in spiking neuromorphic networks with plasticity. Nat Commun 11, 2853 (2020). https://doi.org/10.1038/s41467020165483
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DOI: https://doi.org/10.1038/s41467020165483
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