Neural timescales from a computational perspective
Roxana Zeraati Anna Levina Jakob H. Macke Richard Gao

Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces
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    September 9, 2024
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§ ABSTRACT

Timescales of neural activity are diverse across and within brain areas, and experimental observations suggest that neural timescales reflect information in dynamic environments. However, these observations do not specify how neural timescales are shaped, nor whether particular timescales are necessary for neural computations and brain function. 
Here, we take a complementary perspective and synthesize three directions where computational methods can distill the broad set of empirical observations into quantitative and testable theories:
We review (i) how data analysis methods allow us to capture different timescales of neural dynamics across different recording modalities, (ii) how computational models provide a mechanistic explanation for the emergence of diverse timescales, and (iii) how task-optimized models in machine learning uncover the functional relevance of neural timescales.
This integrative computational approach, combined with empirical findings, would provide a more holistic understanding of how neural timescales capture the relationship between brain structure, dynamics, and behavior.






§ INTRODUCTION



Ongoing neural activity unfolds across a wide range of timescales that are suggested to be related to the large diversity of behavioral timescales and those of the behavior-relevant inputs to the nervous system.
These timescales are often characterized by the decay rate of a neural signal's autocorrelation function. While precise definitions differ, recent experimental studies of neural timescales present a consistent picture across different recording modalities, species, and cognitive tasks: timescales increase along the cortical hierarchy, are diverse within each brain area, and are often correlated with behavioral variables during cognitive tasks (for recent reviews of empirical evidence, see<cit.>). 
As such, this fundamental quantity—the characteristic time constant of neuronal dynamics—may be an important signature of neural circuits across the brain. Importantly, neural timescales are suggested to take on a “representational” role, i.e., the diversity of timescales across the brain mirrors the multiple fluctuation timescales of relevant information in a dynamic environment, which is leveraged by the brain for perception, memory, planning, and action. However, while the large body of experimental observations demonstrates the relevance of neural timescales, they often blur crucial differences between signal modalities, and do not specify the mechanisms underlying how neural timescales arise, nor whether they are necessary (or sufficient) for representation and transformation of information in the brain.


Here, we take a computational perspective and review how models of the brain—when constrained by multi-modal observations of neural timescales—can help provide a mechanistic understanding of neural timescales complementary to experimental investigations. Specifically, we discuss three research directions for how computational methods can distill the broad set of empirical observations into quantitative and testable theories, formulated as the following questions (<ref>):


    
  * “Neural timescales” are measured from different physiological signals (e.g., spikes, field potentials, calcium imaging, fMRI BOLD) and quantified using different methods: how do these measurements relate and differ from one another?
    (Section <ref>)
    
  * Which cellular and circuit processes contribute to the diversity of observed neural timescales, and how can mechanistic models of brain dynamics help propose and exclude candidate mechanisms? (Section <ref>)
    
  * When are “representational neural timescales” that mirror environmental timescales necessary (or sufficient) for supporting task-relevant computations, and how to use functional models from machine learning (in particular, recurrent neural networks) to explore such hypotheses? (Section <ref>)

  

We advocate using an integrative perspective, combining computational and empirical approaches, to understand how neural timescales provide a link between brain structure, dynamics, and behavior and discuss studies that laid stepping stones to build such a perspective.








§ ESTIMATING TIMESCALES FROM NEURAL RECORDINGS





Neural timescales are estimated from different recording modalities, such as spiking activity<cit.>, LFP<cit.>, ECoG<cit.>, calcium  imaging<cit.>, fMRI BOLD<cit.>, using a variety of estimation methods<cit.>. 
Different modalities capture neural activity with distinct spatiotemporal resolutions (e.g., fast and local electrophysiological recordings versus slow and global fMRI BOLD), reflect different underlying physiological mechanisms (e.g., neuronal membrane potential versus metabolic processes), and are described by different mathematical models (e.g., point processes versus continuous signals).
At the same time, estimation methods for corresponding timescales rely on distinct mathematical assumptions that may affect the subsequent results.
While studies of neural timescales often concentrate on the similarity of findings across different modalities and methods, both the recording modality and the analysis method can introduce constraints and biases on estimated timescales that need to be carefully considered when assessing the generalisability of results.







 §.§ The impact of recording modality on estimated timescales

One of the most robust observations across various recording modalities is the hierarchical organization of neural timescales, i.e., timescales being faster in primary sensory and motor areas, and slower in transmodal association areas.
This hierarchy has been observed in spiking activity<cit.>, LFP<cit.>, ECoG<cit.>, calcium  imaging<cit.> and fMRI<cit.>. 
However, despite the similar hierarchical organization, these observations have clear distinctions (<ref>a).


Timescales measured from different modalities are affected by the biophysical properties of recorded signals and physical constraints in the recording setup.
Spiking activity timescales are often in the range of tens to hundreds of milliseconds<cit.>, whereas fMRI BOLD timescales are in the range of a few seconds<cit.>. 
For the same stimulus, the hemodynamic evoked response is generally slower than spiking activity<cit.>.

Moreover, the currently used slow fMRI scan rates may further limit the detection of faster timescales in neural activity<cit.>.
Similar issues can affect timescales measured from calcium imaging:
The time constant of calcium release, as well as the time constant of the calcium indicator and the imaging scan rate, can cause the same underlying neural dynamics to appear as having a different timescale<cit.>. 
Thus, when interpreting their potential functional relevance, it is important to distinguish whether estimated timescale differences arise from underlying population dynamics, biophysics of the measurement process, or both.


On the other hand, neural timescales estimated from different recording modalities may reflect distinct underlying neurophysiological processes and computations that are of further interest.
fMRI BOLD is often associated with population activity <cit.>, hence, its timescales may reflect population spiking timescales.
Estimated timescales from LFP<cit.> and ECoG<cit.>; however, are approximately ten times faster than spiking activity in similar brain regions, which seems to be at odds with the notion that LFP and ECoG signals are slower aggregate signals than spiking activity<cit.>.
This observation suggests that timescales estimated from different modalities may arise from distinct but potentially correlated physiological processes, e.g., fast transmembrane and synaptic currents versus slow recurrent interactions<cit.>.
In fact, when explicitly accounting for multiple timescales in the estimation method<cit.>, both fast and slow timescales can be uncovered from spiking activity<cit.>.
Therefore, estimation methods that consider multiple timescales can potentially incorporate modality- or even behavior-specific time constants explicitly (as we discuss in the next section<cit.>).

The observed hierarchical organization of intrinsic timescales presents differences across studies, potentially stemming from differences across species and methodologies.
Unlike primate studies<cit.>, studies on the hierarchy of timescales in mouse cortex have reported inconsistent results.
Intrinsic timescales measured from calcium imaging<cit.> and fMRI<cit.> display a hierarchical structure from visual to prefrontal areas. 
However, in spiking activity, such a hierarchy was initially observed only in timescales of stimulus-evoked activity <cit.> and not in spontaneous activity <cit.>, though recently it was shown that by analyzing longer periods of recorded neural activity, the hierarchical structure can also be recovered in spontaneous activity<cit.>.
These results suggest that multiple timescales might be present in spontaneous activity, but only some follow the anatomical hierarchy.
Moreover, even in primate studies, the hierarchical ordering of brain areas based on timescales does not exactly match across different modalities<cit.> (<ref>a). 
Studying such subtle but consistent differences may help to better understand the inherent biophysical differences between various recording modalities, and cross-species differences in cortical architecture.





 §.§ Different methods for estimating timescales


Methods for estimating timescales from neural activity usually quantify the decay rate of the signal's autocorrelation function. 
However, each method relies on different theoretical assumptions and incorporates different features of neural dynamics (<ref>b).
For instance, they could explicitly include oscillatory components that are often present in neural activity and can be modulated by task variables, which can alter the autocorrelation shape. Here, we catalog existing methods along two axes: (i) the explicitness of their fitted model and their computational complexity, and (ii) the number of features they capture in data (<ref>b).


The first group of methods does not assume any strict mathematical form for the autocorrelation shape. These methods characterize the decay rate as, for example, the time lag where autocorrelation drops to 1/e<cit.>, the (half of) full width at half maximum<cit.>, or the integral of positive autocorrelation values<cit.>. We thus refer to these methods as “model-free” methods.
Computing such measures is straightforward, and they are typically estimated directly from the data.
However, such model-free methods do not capture the full shape of data autocorrelation. As a result, in the presence of complex temporal features in neural activity beyond a single exponential decay, different methods can give different results on the same data (for example, see <cit.>). 



The second group of methods assumes a mathematical function for the shape of neural data autocorrelation to capture the relevant data features and estimate the timescale from the parameters of the fitted function (i.e., model-based).
The simplest assumption is that the autocorrelation of neural activity follows an exponential decay function, AC(t) = e^-t/τ, where decay time constant τ reflects the timescale of underlying neural dynamics.
This is equivalent to the assumption of obtaining a Lorentzian function as a power spectral density in the frequency domain, i.e., PSD(f) ∼ (f^2 + f_k^2) ^-1, where the knee frequency f_k is translated to timescale as τ = (2π f_k)^-1 <cit.>.
The exponential decay (Lorentzian function) can be further augmented with additional features to better capture the complexity of temporal dynamics.
For example, the autocorrelation of neural dynamics can exhibit more than one timescale<cit.>.
Very slow timescales can also arise from firing rate fluctuations across different trials.
Such a slow timescale can be accounted for by fitting the autocorrelation with an exponential decay function and an additional offset term corresponding to longer-term processes (e.g.,<cit.>, see<cit.> for a toolbox).
Alternatively, multiple timescales can be estimated by fitting the PSD with a mixture of Lorenzian functions<cit.>.


Neural activity can additionally contain oscillatory dynamics that appear as periodic fluctuations in the autocorrelation or peaks in the PSD, which can impact the estimated timescale or knee frequency, respectively.
To overcome this problem, methods such as Spectral Parameterization (or, “FOOOF”) remove the oscillatory components from the PSD before estimating the knee frequency<cit.>. 
For example, oscillatory peaks can be modeled as a mixture of Gaussians and removed. 
Then, the timescale is estimated by fitting an extended form of the Lorentzian function as PSD(f) = C (f^χ + f_k^χ)^-1, with a variable 1/f power-law exponent, allowing a more flexible fitting of the neural power spectra.



Finally, a more accurate estimation of timescales can be obtained using a generative process capturing the statistics of neural time series<cit.>.
Such methods are particularly powerful since they can explicitly incorporate modality-specific properties and task variables.
For instance, including task variables in a multi-timescale autoregressive process allows for disentangling intrinsic and task-induced timescales such as fluctuations induced by reward<cit.>.
Moreover, generative models can capture experimental limitations, such as short trial durations that can lead to systematic underestimation of timescales<cit.>. 


The parameters of the generative process can be directly fitted to neural time series<cit.> (e.g., via least squares or maximum likelihood) or can be used to instantiate a generative model in Bayesian inference methods<cit.>.
Bayesian methods aim to uncover the whole set of parameters that is compatible with the data (i.e., the posterior distribution) instead of obtaining the single best-fitting value of timescales (or other parameters), allowing them to better deal with the stochasticity of neural activity.
For simple models, the posterior distribution can be computed analytically<cit.>, but for more complex models, simulation-based methods such as Approximate Bayesian Computation (e.g., abcTau<cit.>) and simulation-based inference (SBI<cit.>) can be applied to obtain approximate posterior distributions.
In such methods, synthetic time series are produced from an explicit generative (statistical or mechanistic) model mimicking similar statistics as neural recordings. 
Then, by matching the summary statistics (e.g., autocorrelation, PSD, or time series) of synthetic data to the experimental data, we can estimate timescales using the parameters of the matched generative model.

The generative model and summary statistics can be chosen flexibly, allowing them to capture different data features. 
It is possible to fit different generative models (e.g., models with different numbers of timescales or oscillatory components) and use Bayesian model comparison techniques to find the best models describing the neural data<cit.>.
Since the generative model can also incorporate modality-specific properties, such as timescale ranges, stochastic elements (e.g., spiking activity as a point process versus LFP as a continuous signal <cit.>), or even particular biophysical properties, it can be used to uncover possible models describing each modality.
Therefore, these methods can incorporate various mechanistic assumptions and data features when estimating timescales, but this comes with additional computational costs, both in running the simulations and during posterior inference.






§ MECHANISTIC MODELS OF TIMESCALES IN THE BRAIN





Mechanistic models, when constrained by experimental observations of neural timescales, can provide insights into how the diversity of timescales arises within and across different brain areas.
Such models have so far been used to explain the following experimentally observed properties of neural timescales: (i) the hierarchical structure of neural timescales across different brain areas<cit.>, (ii) the diversity of timescales within one brain area<cit.>, and (iii) the flexibility of timescales to change according to different task variables<cit.>.
Mechanisms for explaining the diversity of neural timescales are generally categorized into cellular and synaptic mechanisms on the level of individual neurons and network-mediated mechanisms through interactions between neurons (<ref>). 
While these mechanisms define the “intrinsic” timescales of neurons, other processes, such as external input and neuromodulatory mechanisms, can dynamically modulate them in addition.
Moreover, different models capture neural timescales on different spatial scales of observations, e.g., timescales of single neurons versus population activity.
Thus, comparing these models can reveal how neural timescales across different scales and modalities can be related to each other.


[width=,colback=white,title=Box 1. Timescale as an estimated quantity vs. time constant as a parameter,colbacktitle=babyblueeyes,coltitle=black]
Neural timescales in the literature, for the most part, refer to the estimated decay time constant of measured neural activity, i.e., by describing the autocorrelation function of spike counts or LFP time series as ACF(t)=e^t/τ_fit. Methods discussed in Section <ref> generally aim to find the value of τ_fit (or multiple τ_fits) that best fit the data.


On the other hand, mechanistic (i.e. Hodgkin-Huxley) and functional (i.e. rate-based artificial recurrent neural networks) neural models in Sections <ref> and <ref> typically include a time constant parameter that describes how the impact of input into a neuron decays over time, e.g., τ_paramdV/dt = -(V(t)-V_rest) + I(t), where I(t) includes time-varying stimulus-driven, feed-forward, and recurrent inputs.


It is important to note that the former, estimated τ_fit does not necessarily need to correspond to the latter, underlying model parameter τ_param, and in general, do not. Section <ref> outlines multiple mechanisms, including non-time constant related ones such as network connectivity, that contribute to shaping τ_fit we observe in data.









 §.§ Cellular and synaptic mechanisms


In computational models, timescales are often controlled by incorporating various biophysical time constants of cellular and synaptic processes in the dynamics of individual neurons that reflect heterogeneous properties of ionic channels and synapses<cit.>.
Most dynamical system models include a time constant in the dynamical equation of each neuron, interpreted as the membrane time constant (5-50 ms<cit.>) (Box 1).
Slower timescales can be implemented by introducing other biophysical features such as slow synaptic time constants<cit.>, timescales of short-term synaptic depression and facilitation <cit.>, or adaptation mechanisms<cit.>.
Spike-frequency adaptation, in particular, spans a wide range of fast and slow timescales following a power law, which is suggested to lead to efficient coding in individual cortical neurons driven by behaviorally relevant signals<cit.> and facilitate computations on temporally dispersed inputs <cit.>.





However, biophysical time constants at the single-neuron level cannot explain all the temporal structures of neural dynamics.
For instance, a Hodgkin-Huxley model of an isolated 
neuron with detailed dynamics of membrane potential and ionic and synaptic currents cannot capture the full autocorrelation shape of neural activity recorded in prefrontal areas<cit.>.
Biophysical time constants alone define the autocorrelation timescale of neural activity—henceforth referred to as activity timescale (Box 1)—when a neuron is isolated, but when neurons are embedded in a network, their activity timescale may differ significantly from their biophysical time constants.
At the same time, distinct biophysical mechanisms may interact differently in the presence of network interactions compared to isolated neurons. In particular, while activity timescales of network-embedded neurons increase with the synaptic time constant<cit.>, their relationship to the adaptation time constant is less straightforward: 
Analyzing similar models of randomly connected excitatory and inhibitory units, Beiran and Ostojic<cit.> reported that activity autocorrelation timescales do not depend on the adaption time constant, whereas Muscinelli et al.<cit.> report a positive relationship.
It is important to note, however, that the autocorrelation of neural activity in networks with adaptation does not follow a simple exponential decay. One possible explanation of the apparent disagreement in results could lie in the different methods used to measure timescales in each study, thus highlighting the need to understand how timescale estimation methods impact the conclusions we draw from empirical data and computational simulations.  
In the next section, we review how incorporating network mechanisms, in addition to cellular and synaptic mechanisms, could help to better explain observations of diverse and longer neural timescales.





 §.§ Network and connectivity mechanisms

Neurons in the brain are embedded within rich network structures<cit.> that affect the correlation patterns of neural dynamics<cit.>.
Therefore, a mechanistic understanding of neural timescales may require the context of interactions within a network.
Indeed, computational models have shown that the strength and structure of network connectivity can influence the autocorrelation timescale of neural activity. 
Slower timescales arise from stronger<cit.> or more clustered (e.g., <cit.>) connectivity (for a review see <cit.>).
Strong recurrent connectivity in rate networks shifts the network's dynamical state closer to a critical transition, increasing neural timescales
<cit.>.
In spiking networks, clustered connectivity creates metastable dynamics, and noise-induced transitions between these states give rise to slow timescales<cit.>.
Metastable dynamics can also emerge in homeostatically regulated spiking networks<cit.>.
Such structured connectivity leads to multiple dominant timescales in local dynamics, creating autocorrelation shapes beyond a single exponential decay<cit.>.
Each timescale is created by neural interactions on different spatial scales, such that slower timescales arise from interactions on a larger spatial scale, consistent with similar observations and theories on the spatiotemporal scale of oscillatory brain activity<cit.>.
Introducing heterogeneity in connectivity strengths gives rise to distinct timescales in each network unit<cit.>. 
In addition to recurrent interactions within one brain area, long-range connections between areas are essential for explaining the observed hierarchy of timescales<cit.>. 


Recurrent inhibition also plays an important part in forming network-mediated neural timescales.
Strong inhibitory-to-inhibitory connections allow for slow neural timescales required for working memory maintenance in a recurrent network model<cit.>.

Moreover, in a detailed biophysical model of excitatory and inhibitory neurons, inhibition controls the stability and transitions of metastable states, giving rise to slow timescales<cit.>.
In this model, the full shape of neural autocorrelations in frontal areas is explained by a combination of cellular (i.e., after-hyperpolarization potassium and inhibitory GABA-B conductance) and network mechanisms.
In addition, recurrent inhibition together with membrane and synaptic time constants can account for distinct temporal dynamics in spiking activity and LFP<cit.>.


Overall, computational modeling results have implicated a combination of synaptic, cellular, and network mechanisms to be partially responsible for the emergence of diverse neural timescales in the brain and across different modalities. 
While some of these mechanisms have found empirical support, either via testing different model predictions with experiments<cit.> or
through direct measurements and perturbations of the relevant physiological properties<cit.>, 
 and indirect and post-hoc correlations of neural timescales against, e.g., expression of ion channel and synaptic receptor genes (e.g.,<cit.>), continued interaction between computational predictions and experimental measurements will further shed light on the mechanistic factors underlying variations in intrinsic neural timescales.





 §.§ Modulatory mechanisms for dynamically shaping timescales
 

The mechanisms discussed so far explain how a specific range of timescales can emerge in neural dynamics due to “static" properties of single neurons and networks. 
However, neural timescales are functionally dynamic, and have been shown to adapt to task requirements<cit.>.
Here, we discuss how neuromodulatory processes, changes to feedforward and feedback inputs, and other processes can modulate cellular and network properties, and therefore the resulting activity timescales we measure.

Neuromodulators can impact the biophysical properties of individual neurons (e.g., the response time course) and their network interactions (e.g., gain modulation) (see <cit.> for more extended reviews of the relation between the two and their impact on cortical computation), both of which can alter neural timescales.
Synaptic time constants can be altered by neuromodulators such as acetylcholine (ACh)<cit.>, which can also modify the efficacy of synaptic interactions across the brain in a selective manner<cit.>. 
An increase in ACh strengthens the thalamocortical synaptic efficacy by affecting nicotinic receptors.
At the same time, it weakens lateral cortical interactions while increasing the inhibitory drive by affecting muscarinic receptors.
Similarly, the different contribution of glutamatergic receptors (e.g., NMDA and AMPA) in feedforward and recurrent pathways affects the time course of neural population responses in distinct ways<cit.>. 

External (e.g., sensory) inputs to a network can also modulate neural timescales through a variety of mechanisms.
Input can bring individual neurons to a high-conductance state (lower resistance), which in turn reduces the membrane time constant<cit.> facilitating the distinction of distant synaptic inputs (for a review, see <cit.>).
Moreover, in the presence of non-linear recurrent interactions, the input can modulate the global state of network dynamics<cit.> and alter activity timescales<cit.>. 
The spatial and temporal scales (e.g., size and duration) of the stimulus can also distinctly impact fast and slow neural timescales. 
As a simple example, the activity of a network that tracks external signals over time will trivially exhibit the same timescales.
Furthermore, the spatial extent of stimuli amplifies correlations at fast timescales while reducing them at slow timescales<cit.>.  
In spiking networks, populations with fast (slow) intrinsic timescales respond stronger to faster (slower) components of the input<cit.>.
Thus, depending on the spatial or temporal profile of the input, different timescales might dominate the recorded neural activity. 


The models discussed in this section suggest potential mechanisms underlying the diversity and flexibility of neural timescales.
However, for these models to be more faithful representations of neural circuits, they need to reproduce not only the experimentally observed timescales but also capture other aspects of neural dynamics (e.g., realistic firing rates).
Indeed, considering additional constraints can help to dissociate between various mechanisms that can explain the same range of neural timescales<cit.>.
In addition, constraining computational models that were not specifically designed to explain neural timescales with realistic timescale measurements can provide new insights into different circuit mechanisms.
For instance, constraining models to produce realistic timescales has been instrumental in developing models explaining the mechanisms underlying noise correlations observed in spiking activity<cit.>.
Given the relevance of neural timescales for brain computations and behavior, checking for realistic timescales (e.g., similar to realistic neural firing rates) in computational models can be used as a test for the plausibility of proposed models, and to contrast different mechanistic models which give rise to similar dynamical features, as we discuss in the next section. 






§ COMPUTATIONAL BENEFITS OF THE DIVERSITY AND FLEXIBILITY OF TIMESCALES





As previously mentioned, recent experimental studies have consistently observed that timescales of both single-unit and population dynamics correlate with variables such as behavioral timescales and task performance, thus implicating neural timescales to be a signature of dynamical properties necessary for relevant computations <cit.>. 
However, to make causal claims about the necessity and sufficiency of dynamics operating on different timescales for network computation, perturbation experiments are required to assess the functional impact of timescales over and above, e.g., firing rate modulations, which can be challenging in vivo. 


Recently, task-performing artificial neural network (ANN) models offer a viable, complementary direction that circumvents the practical challenges of manipulating timescale-related variables in the in vivo brain, while enabling us to probe loss- or gain-of-function in experimental task paradigms. 
In computational and cognitive neuroscience, these networks are typically designed to mimic the input-output behavior observed in animal or human participants performing a task<cit.>. 
To match the target output, the parameters of ANNs are either “hand-crafted” when possible (e.g., for simple tasks) or “trained” using an optimization algorithm minimizing a loss function (e.g., for larger networks and more complex tasks).
The networks can then be further analyzed and perturbed to generate hypotheses for the mechanisms of computation at hand (as reviewed in, e.g., <cit.>). 


As a result, ANNs have been used to ask whether mechanisms shaping dynamics, such as connectivity, are important for computation, and whether they are preserved between artificial and biological networks performing the same task.
Various types of ANNs—including spiking neural networks (SNN), rate-based recurrent neural networks (RNNs), and deep learning networks (DL)—can be leveraged to perform task-relevant computations while we explicitly measure, optimize for, and/or perturb their single-neuron and network timescales (Fig. <ref>a). 


In this section, we review two broad categories of timescale-related investigations using ANNs: First, we discuss those that study neural timescales as an emergent characteristic of task-performing networks, including both recent ones employing task-optimized networks, as well as classical works on dynamical properties of “hand-tuned” networks. Second, we discuss recent studies that explicitly optimize timescale-relevant parameters to enhance computational capacity in artificial networks, including biologically plausible spiking and rate networks, and more abstract models in the context of machine learning.






 §.§ Emergent timescales in task-performing neural networks



Computational neuroscience has had a long-standing interest in the repertoire of available timescales to elucidate how neural circuits can represent and perform cognitive tasks with complex temporal structures.
Classically, models were “hand-crafted”, i.e., parameters were manually selected or analytically determined to satisfy a priori constraints, and were intended to capture how neuronal circuits implement dynamics necessary for computation, such as delayed persistent activity for working memory<cit.> or slow evidence accumulation for decision-making<cit.>.
To understand the underlying mechanisms, related theoretical investigations focused on how tailored structures in recurrent network connectivity and its eigenspectrum shape the dynamical landscape or phase portrait of the network<cit.> (Box 2), with special interests in, e.g., fixed points, line attractors, and limit cycles (Fig. <ref>b), which could be leveraged for memory<cit.>, input integration<cit.>, and timing<cit.> respectively (see <cit.> for a recent review on attractor models of neural computation).
Crucially, the timescales of the latent dynamical system near and on these low-dimensional manifolds are emergent properties of interest, since they govern, for example, how quickly and how stably a network can integrate new input while remembering or forgetting old ones, filter out fast noisy fluctuations to sense slower signals, and receive inputs of different frequencies.


Concurrently, related research on reservoir computing, including liquid state machines <cit.> and echo state networks <cit.>, similarly considered the activity timescales of RNNs from the perspective of latent dynamical systems, since they require long-term but not divergent (or exploding) dynamics at the edge-of-chaos <cit.>.
While reservoir networks typically have trained input and output weights, the recurrent weights are fixed due to the challenges of propagating error over many time steps, and much attention was dedicated to investigating how they can be thoughtfully initialized to maximize the network's computational capacity (similar to <cit.>).
Particularly relevant was how input-driven chaos in such networks can be tuned to represent a flexible repertoire of timescales (or temporal filters) for time-dependent computations <cit.>.


[width=,colback=white,title=Box 2. Activity timescales of a linear network,colbacktitle=babyblueeyes,coltitle=black, float, floatplacement=t]
As an illustrative example, we review how to calculate the activity timescale (τ_fit) of a linear recurrent network (i.e., linear dynamical system). Consider the system

    dx(t)/dt = Ax(t)

where x(t) is the state vector, e.g., neural activity at time t, and A is a square connectivity matrix.
To acquire the timescales of the system dynamics, we solve for the eigenvalues of the matrix A, λ_i, through the characteristic equation, Au_i = λ_i u_i. 


For a stable decaying mode, i.e., non-exploding, the real part of the associated eigenvalue must be negative, and its decay timescale is defined as

    τ_i = -1/Re(λ_i).

The dominant activity timescale of the system (i.e., τ_fit) is then typically determined by the mode whose eigenvalue has the largest real part.


Furthermore, for a linear network composed of units with fixed membrane time constant τ_param, the system dynamics follow

    τ_paramdx(t)/dt = Ax(t).

Assuming τ_param to be the same for all neurons, the activity timescale is scaled accordingly,

    τ_i = -τ_param/Re(λ_i).

Extensions to networks with further biological constraints, e.g., random networks obeying Dale's law, can also be found <cit.>.




These earlier studies often considered task constraints from a theoretical and analytical perspective <cit.> (Fig. <ref>c, left), and they motivated recent investigations using task-optimized networks to tackle similar questions in more complex or analytically intractable networks.
The advancement of modern machine learning hardware accelerated network optimization strategies for RNNs (e.g., backpropagation-through-time <cit.> via stochastic gradient descent) and enabled a complementary approach:
instead of handcrafting networks with certain dynamical properties required for a task, the recurrent weights and other parameters of networks can be optimized directly to perform tasks or to mimic experimental recordings <cit.> (Fig. <ref>c, middle and right), and subsequently analyzed to identify the repertoire of neural dynamics important for the required computation (i.e., “opening the black box"<cit.>).
Consistent with earlier works on hand-crafted networks, analyses often reveal fixed points corresponding to different inputs or memories <cit.>, line attractors for tracking continuous environmental or behavioral variables <cit.>, and (quasi-)limit cycles or state space trajectories traversed at different speeds that correspond to the timing of network inputs or outputs <cit.>, while multi-task networks exhibit context- and task phase-dependent composition of the above “computational primitives” <cit.>.


In particular, efficient optimization of computational models has opened up recent investigations into the connection between observed neural activity and the underlying latent dynamical system.
By fitting models of neural population activity with latent states governed by, for example, switching linear dynamics, one could extract the time constants of dynamical objects of interest, such as approximate line attractors representing evidence or internal state accumulation <cit.>.
Moreover, one could directly compare the temporal dynamics of individual neurons' activity in experimental measurements with that of neurons in task-performing network models (Fig. <ref>d), both in rate-based and spiking recurrent neural networks, to dissect cellular, network, and learning-related mechanisms of computations underlying e.g., temporal scaling <cit.>, working memory <cit.>, and other motor or cognitive tasks <cit.>. 


These ideas can be further extended to study networks endowed with more sophisticated and biologically-realistic constraints, such as Dale's law<cit.> and dendritic compartmentalization <cit.>, hypothesized neural computational mechanisms such as oscillatory phase coding<cit.>, as well as better analytical tractability (e.g., low-rank RNNs<cit.>) or improved expressiveness, computational power, and trainability <cit.>). Targeting the same sought-after dynamical repertoires, “hybrid” approaches can leverage more sophisticated weight initialization or regularization strategies to nudge the network toward line attractors or limit cycles after training <cit.>. Furthermore, since population activity timescales reflect the state of neural dynamics relative to critical transitions, timescales have also been used to study how the state of dynamics in spiking neuromorphic networks affect the performance of tasks with varying complexity<cit.>. 


Altogether, these studies provide insights into specific tasks or network architectures underlying computations of interest from the perspective of network dynamics. Leaning on the notion that the geometry of neural population representation is intimately linked to (the timescales of) its dynamics and that the latter may be even more important than the former <cit.>, both classical and recent works lend support to the importance of the diversity of neural timescales for different computations. In other words, the temporal signatures of a dynamical system are a reflection of its phase portrait geometry, and constraining one to satisfy certain computational requirements often means constraining the other as well<cit.>. Nevertheless, it may be possible for networks to acquire similar low-dimensional geometries but vastly different timescales of dynamics. Therefore, we suggest explicit comparisons of in silico and in vivo timescales in task-performing networks as a straightforward yet synergistic research direction to provide explicit evidence for the necessity (or the lack thereof) of different neural timescales.




 §.§ Optimization of timescale-related parameters


Another powerful way to use modern ANNs to study the effect of neural timescales on computation is to optimize timescale-related parameters in the network model, such as synaptic, membrane, and adaptation time constants, or more phenomenological variables, such as temporal receptive field size and context length. In a standard RNN, all neurons are given a single fixed time constant that is roughly analogous to the membrane time constant of biological neurons, which determines how quickly the effect of an input on that neuron decays. In reality, neurons across the brain have a range of membrane time constants due to varying properties such as dendritic morphology and the composition of conductances (see Section <ref>). As such, time constants can be individually optimized (similar to the connectivity weight matrices) <cit.>, or even parameterized to be input-dependent, arriving at gated network architectures (e.g., long short-term memory units, or LSTMs). Thus, by comparing to networks that are not endowed with diverse timescale parameters, such studies provide evidence for the potential necessity of different timescales in behaviorally relevant computations. 


Several recent studies have shown that spiking networks trained to perform standard machine learning or cognitive tasks (e.g., delayed classification, working memory maintenance, etc.) can benefit from both the heterogeneity and explicit optimization of single-neuron or spike-frequency adaptation time constants <cit.>. Strikingly, the trained networks exhibit similar timescale distributions as membrane time constants<cit.> or activity timescales of individual neurons in analogous brain regions of animals performing the same task<cit.>. Furthermore, time constants do not necessarily need to match that of the task, but both longer and diverse time constants were found to be beneficial <cit.>. Similarly, RNNs with trainable time constants demonstrate improved task performance and memory capacity while the learned time constants mimic the hierarchy of timescales in the data-generating process<cit.>. With more bio-realistic RNNs, such as those with connectivity based on the measured connectome of the fruit fly visual system, training both the connection weights and single-neuron time constants resulted in emergent stimulus tuning similar to that of in vivo recorded neurons in the fly<cit.>. 

Remarkably, multi-layer feed-forward deep neural networks trained to simply mimic the input-output relationship of a single morphologically accurate pyramidal neuron recover complex and spatially varying temporal filters <cit.>, further emphasizing the computational capacity provided by multiple timescales, even at the scale of single neurons. Recurrent architecture fitted to the same morphologically accurate pyramidal neuron allows for the explicit implementation of multiple timescales within a neuron's model and enables a single neuron with capabilities to solve complex long temporal dependencies tasks <cit.>.  
On the other hand, diverse activity timescales can arise from network interactions instead of cellular properties (as discussed in the previous section). A recent study suggests that developing network-mediated diverse and long timescales may be a more robust mechanism for performing long memory tasks rather than trainable time constants <cit.>. Hence, future studies need to further distill the computational role of diverse cellular or network-mediated timescales.


These principles extend to more complex deep learning-based network architectures, providing functional benefits on more difficult tasks. Instead of trainable and fixed time constants, they can be further made to be input-dependent via trained gating weights (similar to gated RNNs in DL, such as GRUs and LSTMs), which improves modeling of sequences with multiple timescales <cit.>. Furthermore, initializing gating parameters to reflect the distribution of timescales in the data improves performance on tasks involving memorizing and operating on long sequences <cit.>. Leveraging this observation, endowing LSTM units with a distribution of effective timescales constructed to match the power-law temporal dependency of natural language improves network performance in language modeling, especially long-range dependencies <cit.>. In parallel, these “chrono initialized" LSTMs improved decoding of brain response to narrative text, as well as estimation of voxel-wise fMRI BOLD timescales along the cortical hierarchy in human subjects<cit.>, uniting potential principles of operation underlying both artificial and biological neural networks. 

Finally, recent research in deep learning has shown an intriguing convergence of principles based on thoughtful inductive biases and initialization, as well as self-supervised pre-training, for improving performance on tasks with long-range dependencies <cit.>.
Although RNN-based architectures can model infinitely long temporal dependencies in theory (similar to IIR filters), simpler implementations based on finite-length temporal convolutions appeared to be more effective at acquiring long timescale memories <cit.>, though both architectures learn effective temporal context windows that reflect natural structures in data (e.g., speech <cit.>).
More recent architectures based on deep (linear) state space models (SSMs, e.g., S4 <cit.> and others <cit.>) enjoy the benefits of both recurrence and convolutions, outperforming Transformer-based architectures on long timescale tasks, which are hindered by quadratic complexity in time.
Later extensions capitalize on these insights to derive better parameterization or initialization strategies, resulting in competitive models with simpler implementations, such as Structure Global Convolution <cit.> and Linear Recurrence Units <cit.>.
On the other hand, it has been shown that both Transformer-based and SSM-based architectures can be drastically improved through self-supervised pre-training on the input sequence data, prior to downstream task training (e.g., labeled classification) on the same long-range tasks <cit.>.
These findings suggest that equipping deep learning models with inductive biases appropriate for long timescale tasks can be promising, but the exact mechanisms of these improvements—and whether they are related to intrinsic timescales in biological networks—remain unclear.

In summary, there is a growing interest in convergent approaches to tuning network models to produce emergent task-relevant dynamics and timescales, as well as explicit optimization of time-constant parameters in ANNs. These approaches appear to be a promising and collaborative avenue for research in neuroscience and machine learning aiming to understand and improve time-dependent computations in biological and artificial systems.





§ DISCUSSION




In this paper, we discussed how different computational methods help to turn empirical observation of diverse neural timescales into quantitative and testable hypotheses about brain dynamics:
Different timescale estimation methods can disentangle various components of neural activity, such as modality- or task-specific variables, from intrinsic dynamics, while
mechanistic models constrained by experimentally observed timescales, together with task-performing ANNs, can elucidate potential circuit mechanisms underlying brain dynamics and computations. 
Integrating these directions, we here provide an outlook for future research and discuss open questions.




 §.§ Methodological suggestions


Given the differences in assumptions and applicability of the various timescale estimation methods we outlined, it is important to be aware of their limitations when interpreting consistencies and differences in existing results, as well as when designing future methodological and empirical research questions. Since there is not one single correct method, it would be ideal if each dataset was analyzed using all possible methods—a decidedly unrealistic proposal. Short of that, a systematic and quantitative comparison of different timescale estimation methods would serve as an extremely useful reference for contextualizing current and new results, while guiding the development of novel methods that address shared shortcomings of existing methods. Complementary to method development, neural timescale can be a standardized metric for characterizing neural dynamics at multiple scales, for which a curated database of existing estimates across brain regions, recording modalities, species, and task conditions—similar to databases of single-neuron electrophysiological measurements <cit.>—would be invaluable.




 §.§ Models and data, an inseparable loop

Computational models suggest that various cellular and network mechanisms could alter timescales in the brain. Therefore, timescale-constrained models can be used as a great tool to provide mechanistic insights into brain dynamics and computations.
It is important to note again that activity timescales of neurons may differ significantly from time-constant parameters implemented in the models (e.g., membrane or synaptic time constants) (Box 1).
Hence, it is crucial to match the activity timescales of the models to those of the data.
For models with fewer mechanisms or free parameters, it is possible to derive the activity timescales analytically<cit.>.
However, in the brain, multiple mechanisms interact together to shape timescales, and studying their interactions analytically is not straightforward.
In such cases, qualitative matching, e.g., generating a similar order of hierarchy of timescales<cit.> or a grid search over multiple model parameters<cit.> are used to fit the model timescales to data.


Nevertheless, such approaches may be inadequate when the parameter space is high-dimensional and the interaction between parameters is unknown. Fitting detailed mechanistic models with high-dimensional parameter space is possible using optimization or inference procedures such as genetic algorithms<cit.>, Approximate Bayesian Computation (ABC)<cit.> and simulation-based inference (SBI)<cit.>.
In particular, since ABC and SBI estimate the posterior distribution of parameters, they identify degeneracies in parameter space and uncover potential relations between different parameters, which helps us better understand how combinations of mechanisms can give rise to observed neural timescales.
Even with often imperfect (or, misspecified) models in neuroscience, when combined with inference methods that appropriately account for such errors<cit.>, fitting mechanistic models to neural activity across multiple modalities<cit.> could provide a holistic understanding of neural timescales across different scales of brain dynamics and may lead to new predictions that can be tested experimentally.



 §.§ Timescales seed further synergies between neuroscience and ML

State-of-the-art machine learning tools have been readily adopted to interpret neural and behavioral data over multiple timescales. Conversely, the works discussed in Section <ref> further attempt to build a theoretical bridge between principles of dynamic neural computation and deep and/or recurrent artificial neural networks. Here, we make two broad remarks that can guide future research.


First, viewing network timescale as the temporal signature of the geometry of its dynamical landscape, it stands to reason that optimizing the network to achieve a certain task-conducive geometry (e.g., line attractor or limit cycle) may be reformulated as optimization for a certain repertoire or diversity of timescales, as recent works have demonstrated a relationship between a network's trainability to the stability of its dynamics (i.e., Lyapunov exponent)<cit.>. This has implications for RNN-based machine learning models, as “pretraining" for certain measures of network dynamic can be done without explicit task instructions<cit.>, and can also provide potential insights for how neural circuits organize themselves during early development without input<cit.>.


Second, there is a growing interest in the emerging field of NeuroAI to develop models that not only can perform cognitive and behavioral tasks, but also produce realistic neural activity<cit.>. While developing such models may seem challenging, matching models' activity timescales to that of neural data can serve as a first step in building task-optimized models with realistic activity fluctuations. Timescale constraints may further alter the solution landscape of such models to what can be more relevant for understanding brain computations. 


At the same time, modern ML models such as large language models or deep reinforcement learning models can facilitate studying the role of neural timescales in naturalistic tasks occurring over longer timescales than what can be studied in common laboratory-constrained experiments. Indeed, a recent study on freely behaving animals reported a different organization of neural timescales across brain areas than what has been found during resting state<cit.>. Therefore, timescales can serve as a key to open new avenues where ML tools can expedite uncovering mechanisms underlying brain computation. 



 §.§ Overlooked timescales and outlook

Much of systems and cognitive neuroscience has been based on a specific set of tasks, often with a trial structure between 1 to 10 seconds long, which artificially limits observed neural and behavioral timescale to a small range of the spectrum. For instance, spontaneous animal behavior in natural environments exhibits correlations over a long range of timescales that cannot be measured in common experimental settings <cit.>. On the other hand, processes such as the release of synaptic vesicles occur on a sub-second timescale <cit.> that are not detectable with many measurement tools. Furthermore, different physiological processes may be involved in shaping neural timescales that are often ignored when studying the mechanisms underlying neural timescales. For example, astrocytes have slow intracellular calcium fluctuations that may mediate slow neural modulations <cit.>. Astrocytes may also play a significant role in neural computations, as recently was shown that neuron–astrocyte networks can perform similar computations to complex machine learning models like Transformers <cit.>.
How processes across different timescales interact may be of tremendous interest: does sub-second dynamics of synaptic vesicles play a role in shaping how, for example, memory engrams are formed over days and months, beyond the mean activity? On the other hand, do longer timescale fluctuations beyond a designated static state like age or disease conditions, such as circadian or seasonal rhythms, impact moment-to-moment neural dynamics? To investigate these questions, we likely need to leverage a combination of different experimental paradigms in the lab and naturalistic settings <cit.>, together with long-term and high temporal resolution neural recording <cit.>, as well as novel data analysis and modeling tools to parse the interaction between and across the different orders of magnitude timescales.




 §.§ Acknowlegments

This work was supported by the German Research Foundation (DFG) through Germany’s Excellence Strategy (EXC-Number 2064/1, PN 390727645) (RZ, JHM, RG) and SFB1233 (PN 276693517) (JHM),  the German Federal Ministry of Education and Research (Tübingen AI Center, FKZ: 01IS18039) (AL, JHM, RG), the Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation endowed by the Federal Ministry of Education and Research (RZ, AL), the European Union (ERC, “DeepCoMechTome”, ref. 101089288) (JHM), the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 101030918 (AutoMIND) (RG).

The authors thank Ana Manea and Jan Zimmermann for providing the timescales of fMRI data (<ref>a), Bradley Voytek and the speakers and participants of the Cosyne 2022 workshop on https://www.rdgao.com/blog/2022/03/28/“Mechanisms, functions, and methods for diversity of neuronal and network timescales" for valuable discussions, 
and https://scidraw.io/Scidraw, https://en.wikipedia.org/wiki/Dopaminergic_pathways#/media/File:Dopaminergic_pathways.svgWikipedia and Pixel perfect from https://www.flaticon.com/Flaticon for providing the basis icons for some of the illustrations.


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