Decision letter: Nonlinear transient amplification in recurrent neural networks with short-term plasticity

Abstract

Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Materials and methods Appendix 1 Appendix 2 Appendix 3 Appendix 4 Appendix 5 Data availability References Decision letter Author response Article and author information Metrics Abstract To rapidly process information, neural circuits have to amplify specific activity patterns transiently. How the brain performs this nonlinear operation remains elusive. Hebbian assemblies are one possibility whereby strong recurrent excitatory connections boost neuronal activity. However, such Hebbian amplification is often associated with dynamical slowing of network dynamics, non-transient attractor states, and pathological run-away activity. Feedback inhibition can alleviate these effects but typically linearizes responses and reduces amplification gain. Here, we study nonlinear transient amplification (NTA), a plausible alternative mechanism that reconciles strong recurrent excitation with rapid amplification while avoiding the above issues. NTA has two distinct temporal phases. Initially, positive feedback excitation selectively amplifies inputs that exceed a critical threshold. Subsequently, short-term plasticity quenches the run-away dynamics into an inhibition-stabilized network state. By characterizing NTA in supralinear network models, we establish that the resulting onset transients are stimulus selective and well-suited for speedy information processing. Further, we find that excitatory-inhibitory co-tuning widens the parameter regime in which NTA is possible in the absence of persistent activity. In summary, NTA provides a parsimonious explanation for how excitatory-inhibitory co-tuning and short-term plasticity collaborate in recurrent networks to achieve transient amplification. Editor's evaluation Many brain circuits, particularly those found in mammalian sensory cortices, need to respond rapidly to stimuli while at the same time avoiding pathological, runaway excitation. Over several years, many theoretical studies have attempted to explain how cortical circuits achieve these goals through interactions between inhibitory and excitatory cells. This study adds to this literature by showing how synaptic short-term depression can stabilise strong positive feedback in a circuit under a variety of plausible scenarios, allowing strong, rapid and stimulus-specific responses. https://doi.org/10.7554/eLife.71263.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Perception in the brain is reliable and strikingly fast. Recognizing a familiar face or locating an animal in a picture only takes a split second (Thorpe et al., 1996). This pace of processing is truly remarkable since it involves several recurrently connected brain areas each of which has to selectively amplify or suppress specific signals before propagating them further. This processing is mediated through circuits with several intriguing properties. First, excitatory-inhibitory (EI) currents into individual neurons are commonly correlated in time and co-tuned in stimulus space (Wehr and Zador, 2003; Froemke et al., 2007; Okun and Lampl, 2008; Hennequin et al., 2017; Rupprecht and Friedrich, 2018; Znamenskiy et al., 2018). Second, neural responses to stimulation are shaped through diverse forms of short-term plasticity (STP) (Tsodyks and Markram, 1997; Markram et al., 1998; Zucker and Regehr, 2002; Pala and Petersen, 2015). Finally, mounting evidence suggests that amplification rests on neuronal ensembles with strong recurrent excitation (Marshel et al., 2019; Peron et al., 2020), whereby excitatory neurons with similar tuning preferentially form reciprocal connections (Ko et al., 2011; Cossell et al., 2015). Such predominantly symmetric connectivity between excitatory cells is consistent with the notion of Hebbian cell assemblies (Hebb, 1949), which are considered an essential component of neural circuits and the putative basis of associative memory (Harris, 2005; Josselyn and Tonegawa, 2020). Computationally, Hebbian cell assemblies can amplify specific activity patterns through positive feedback, also referred to as Hebbian amplification. Based on these principles, several studies have shown that Hebbian amplification can drive persistent activity that outlasts a preceding stimulus (Hopfield, 1982; Amit and Brunel, 1997; Yakovlev et al., 1998; Wong and Wang, 2006; Zenke et al., 2015; Gillary et al., 2017), comparable to selective delay activity observed in the prefrontal cortex when animals are engaged in working memory tasks (Funahashi et al., 1989; Romo et al., 1999). However, in most brain areas, evoked responses are transient and sensory neurons typically exhibit pronounced stimulus onset responses, after which the circuit dynamics settle into a low-activity steady-state even when the stimulus is still present (DeWeese et al., 2003; Mazor and Laurent, 2005; Bolding and Franks, 2018). Preventing run-away excitation and multi-stable attractor dynamics in recurrent networks requires powerful and often finely tuned feedback inhibition resulting in EI balance (Amit and Brunel, 1997; Compte et al., 2000; Litwin-Kumar and Doiron, 2012; Ponce-Alvarez et al., 2013; Mazzucato et al., 2019), However, strong feedback inhibition tends to linearize steady-state activity (van Vreeswijk and Sompolinsky, 1996; Baker et al., 2020). Murphy and Miller, 2009 proposed balanced amplification which reconciles transient amplification with strong recurrent excitation by tightly balancing recurrent excitation with strong feedback inhibition (Goldman, 2009; Hennequin et al., 2012; Hennequin et al., 2014; Bondanelli and Ostojic, 2020; Gillett et al., 2020). Importantly, balanced amplification was formulated for linear network models of excitatory and inhibitory neurons. Due to linearity, it intrinsically lacks the ability to nonlinearly amplify stimuli which limits its capabilities for pattern completion and pattern separation. Further, how balanced amplification relates to nonlinear neuronal activation functions and nonlinear synaptic transmission as, for instance, mediated by STP (Tsodyks and Markram, 1997; Markram et al., 1998; Zucker and Regehr, 2002; Pala and Petersen, 2015), remains elusive. This begs the question of whether there are alternative nonlinear amplification mechanisms and how they relate to existing theories of recurrent neural network processing. Here, we address this question by studying an alternative mechanism for the emergence of transient dynamics that relies on recurrent excitation, supralinear neuronal activation functions, and STP. Specifically, we build on the notion of ensemble synchronization in recurrent networks with STP (Loebel and Tsodyks, 2002; Loebel et al., 2007) and study this phenomenon in analytically tractable network models with rectified quadratic activation functions (Ahmadian et al., 2013; Rubin et al., 2015; Hennequin et al., 2018; Kraynyukova and Tchumatchenko, 2018) and STP. We first characterize the conditions under which individual neuronal ensembles with supralinear activation functions and recurrent excitatory connectivity succumb to explosive run-away activity in response to external stimulation. We then show how STP effectively mitigates this instability by re-stabilizing ensemble dynamics in an inhibition-stabilized network (ISN) state, but only after generating a pronounced stimulus-triggered onset transient. We call this mechanism NTA and show that it yields selective onset responses that carry more relevant stimulus information than the subsequent steady-state. Finally, we characterize the functional benefits of inhibitory co-tuning, a feature that is widely observed in the brain (Wehr and Zador, 2003; Froemke et al., 2007; Okun and Lampl, 2008; Rupprecht and Friedrich, 2018) and readily emerges in computational models endowed with activity-dependent plasticity of inhibitory synapses (Vogels et al., 2011). We find that co-tuning prevents persistent attractor states but does not preclude NTA from occurring. Importantly, NTA purports that, following transient amplification, neuronal ensembles settle into a stable ISN state, consistent with recent studies suggesting that inhibition stabilization is a ubiquitous feature of cortical networks (Sanzeni et al., 2020). In summary, our work indicates that NTA is ideally suited to amplify stimuli rapidly through the interaction of strong recurrent excitation with STP. Results To understand the emergence of transient responses in recurrent neural networks, we studied rate-based population models with a supralinear, power law input-output function (Figure 1A and B; Ahmadian et al., 2013; Hennequin et al., 2018), which captures essential aspects of neuronal activation (Priebe et al., 2004), while also being analytically tractable. We first considered an isolated neuronal ensemble consisting of one excitatory (E) and one inhibitory (I) population (Figure 1A). Figure 1 with 1 supplement see all Download asset Open asset Neuronal ensembles nonlinearly amplify inputs above a critical threshold. (A) Schematic of the recurrent ensemble model consisting of an excitatory (blue) and an inhibitory population (red). (B) Supralinear input-output function given by a rectified power law with exponent α=2. (C) Firing rates of the excitatory (blue) and inhibitory population (red) in response to external stimulation during the interval from 2 to 4 s (gray bar). The stimulation was implemented by temporarily increasing the input gE. (D) Phase portrait of the system before stimulation (left; C orange) and during stimulation (right; C green). (E) Characteristic function F⁢(z) for varying input strength gE. Note that the function loses its zero crossings, which correspond to fixed points of the system for increasing external input. (F) Heat map showing the evoked firing rate of the excitatory population for different parameter combinations JE⁢E and gE. The gray region corresponds to the parameter regime with unstable dynamics. The dynamics of this network are given by (1) τE⁢d⁢rEd⁢t=-rE+[JE⁢E⁢rE-JE⁢I⁢rI+gE]+αE , (2) τI⁢d⁢rId⁢t=-rI+[JI⁢E⁢rE-JI⁢I⁢rI+gI]+αI , where rE and rI are the firing rates of the excitatory and inhibitory population, τE and τI represent the corresponding time constants, JX⁢Y denotes the synaptic strength from the population Y to the population X, where X,Y∈{E,I}, gE and gI are the external inputs to the respective populations. Finally, αE and αI, the exponents of the respective input-output functions, are fixed at two unless mentioned otherwise. For ease of notation, we further define the weight matrix J of the compound system as follows: (3) J=[JEE−JEIJIE −JII]. We were specifically interested in networks with strong recurrent excitation that can generate positive feedback dynamics in response to external inputs gE. Therefore, we studied networks with (4) det(J)=−JEEJII+JIEJEI<0. In contrast, networks in which recurrent excitation is met by strong feedback inhibition such that det(J)>0 are unable to generate positive feedback dynamics provided that inhibition is fast enough (Ahmadian et al., 2013). Importantly, we assumed that most inhibition originates from recurrent connections (Franks et al., 2011; Large et al., 2016) and, hence, we kept the input to the inhibitory population gI fixed unless mentioned otherwise. Nonlinear amplification of inputs above a critical threshold We initialized the network in a stable low-activity state in the absence of external stimulation, consistent with spontaneous activity in cortical networks (Figure 1C). However, an input gE of sufficient strength, destabilized the network (Figure 1C). Importantly, this behavior is distinct from linear network models in which the network stability is independent of inputs (Materials and methods). The transition from stable to unstable dynamics can be understood by examining the phase portrait of the system (Figure 1D). Before stimulation, the system has a stable and an unstable fixed point (Figure 1D, left). However, both fixed points disappear for an input gE above a critical stimulus strength (Figure 1D, right). To further understand the system’s bifurcation structure, we consider the characteristic function (5) F(z)=JEE[z]+αE−JEI[det(J)⋅JEI−1[z]+αE+JEI−1JIIz−JEI−1JIIgE+gI]+αI−z+gE, where z denotes the total current into the excitatory population and det(J) represents the determinant of the weight matrix (Kraynyukova and Tchumatchenko, 2018; Materials and methods). The characteristic function reduces the original two-dimensional system to one dimension, whereby the zero crossings of the characteristic function correspond to the fixed points of the original system (Eq. (1)-(2)). We use this correspondence to visualize how the fixed points of the system change with the input gE. Increasing gE shifts F⁢(z) upwards, which eventually leads to all zero crossings disappearing and the ensuing unstable dynamics (Figure 1E; Materials and methods). Importantly, for any weight matrix J with negative determinant, there exists a critical input gE at which all fixed points disappear (Materials and methods). While for weak recurrent E-to-E connection strength JE⁢E, the transition from stable dynamics to unstable is gradual, in that it happens at higher firing rates (Figure 1F), it becomes more abrupt for stronger JE⁢E. Thus, our analysis demonstrates that individual neuronal ensembles with negative determinant det(J) nonlinearly amplify inputs above a critical threshold by switching from initially stable to unstable dynamics. Short-term plasticity, but not spike-frequency adaptation, can re-stabilize ensemble dynamics Since unstable dynamics are not observed in neurobiology, we wondered whether neuronal spike frequency adaptation (SFA) or STP could re-stabilize the ensemble dynamics while keeping the nonlinear amplification character of the system. Specifically, we considered SFA of excitatory neurons, E-to-E short-term depression (STD), and E-to-I short-term facilitation (STF). We focused on these particular mechanisms because they are ubiquitously observed in the brain. Most pyramidal cells exhibit SFA (Barkai and Hasselmo, 1994) and most synapses show some form of STP (Markram et al., 1998; Zucker and Regehr, 2002; Pala and Petersen, 2015). Moreover, the time scales of these mechanisms are well-matched to typical timescales of perception, ranging from milliseconds to seconds (Tsodyks and Markram, 1997; Fairhall et al., 2001; Pozzorini et al., 2013). When we simulated our model with SFA (Eqs. (21)–(23)), we observed different network behaviors depending on the adaptation strength. When adaptation strength was weak, SFA was unable to stabilize run-away excitation (Figure 2A; Materials and methods). Increasing the adaptation strength eventually prevented run-away excitation, but to give way to oscillatory ensemble activity (Figure 2—figure supplement 1). Finally, we confirmed analytically that SFA cannot stabilize excitatory run-away dynamics at a stable fixed point (Materials and methods). In particular, while the input is present, strong SFA creates a stable limit cycle with associated oscillatory ensemble activity (Figure 2—figure supplement 1; Materials and methods), which was also shown in previous modeling studies (van Vreeswijk and Hansel, 2001), but is not typically observed in sensory systems (DeWeese et al., 2003; Rupprecht and Friedrich, 2018). Figure 2 with 12 supplements see all Download asset Open asset Short-term plasticity, but not spike-frequency adaptation, re-stabilizes ensemble dynamics. (A) Firing rates of the excitatory (blue) and inhibitory population (red) in the presence of spike-frequency adaptation (SFA). During stimulation (gray bar) additional input is injected into the excitatory population. The inset shows a cartoon of how SFA affects spiking neuronal dynamics in response to a step current input. (B) Left: Same as (A) but in the presence of E-to-E short-term depression (STD). Right: Same as left but inactivating inhibition in the period marked in purple. (C) 3D plot of the excitatory activity rE, inhibitory activity rI, and the STD variable x of the network in B left. The orange and green points mark the fixed points before/after and during stimulation. (D) Characteristic function F⁢(z) in networks with E-to-E STD. Different brightness levels correspond to different time points in B left. (E) Same as (B) but in the presence of E-to-I short-term facilitation (STF). (F) Inhibition-stabilized network (ISN) index, which corresponds to the largest real part of the eigenvalues of the Jacobian matrix of the E-E subnetwork with STD, as a function of time for the network with E-to-E STD in B left. For values above zero (dashed line), the ensemble is an ISN. (G) Analytical solution of non-ISN (magenta), ISN (green), paradoxical, and non-paradoxical regions for different parameter combinations JE⁢E and the STD variable x. The solid line separates the non-ISN and ISN regions, whereas the dashed line separates the non-paradoxical and paradoxical regions. (H) The normalized firing rates of the excitatory (blue) and inhibitory population (red) when injecting additional excitatory current into the inhibitory population before stimulation (left; orange bar in B), and during stimulation (right; green bar in B). Initially, the ensemble is in the non-ISN regime and injecting excitatory current into the inhibitory population increases its firing rate. During stimulation, however, the ensemble is an ISN. In this case, excitatory current injection into the inhibitory population results in a reduction of its firing rate, also known as the paradoxical effect. Next, we considered STP, which is capable of saturating the effective neuronal input-output function (Mongillo et al., 2012; Zenke et al., 2015; Eqs. (37)–(39), Eqs. (41)–(43)). We first analyzed the stimulus-evoked network dynamics when we added STD to the recurrent E-to-E connections. Strong depression of synaptic efficacy resulted in a brief onset transient after which the ensemble dynamics quickly settled into a stimulus-evoked steady-state with slightly higher activity than the baseline (Figure 2B, left). After stimulus removal, the ensemble activity returned back to its baseline level (Figure 2B, left; Figure 2C). Notably, the ensemble dynamics settled at a stable steady state with a much higher firing rate, when inhibition was inactivated during stimulus presentation (Figure 2B, right). This shows that STP is capable of creating a stable high-activity fixed point, which is fundamentally different from the SFA dynamics discussed above. This difference in ensemble dynamics can be readily understood by analyzing the stability of the three-dimensional dynamical system (Materials and methods). We can gain a more intuitive understanding by considering self-consistent solutions of the characteristic function F⁢(z). Initially, the ensemble is at the stable low activity fixed point. But the stimulus causes this fixed point to disappear, thus giving way to positive feedback which creates the leading edge of the onset transient (Figure 2B). However, because E-to-E synaptic transmission is rapidly reduced by STD, the curvature of F⁢(z) changes and a stable fixed point is created, thereby allowing excitatory run-away dynamics to terminate and the ensemble dynamics settle into a steady-state at low activity levels (Figure 2D). We found that E-to-I STF leads to similar dynamics (Figure 2E, left; Appendix 1) with the only difference that this configuration requires inhibition for network stability (Figure 2E, right), whereas E-to-E STD stabilizes activity even without inhibition, albeit at physiologically implausibly high activity levels. Importantly, the re-stabilization through either form of STP did not impair an ensemble’s ability to amplify stimuli during the initial onset phase. Crucially, transient amplification in supralinear networks with STP occurs above a critical threshold (Figure 2—figure supplement 2), and requires recurrent excitation JEE to be sufficiently strong (Figure 2—figure supplement 2C, D). To quantify the amplification ability of these networks, we calculated the ratio of the evoked peak firing rate to the input strength, henceforth called the ‘Amplification index’. We found that amplification in STP-stabilized supralinear networks can be orders of magnitude larger than in linear networks with equivalent weights and comparable stabilized supralinear networks (SSNs) without STP (Figure 2—figure supplement 3). We stress that the resulting firing rates are parameter-dependent (Figure 2—figure supplement 4) and their absolute value can be high due to the high temporal precision of the onset peak and its short duration. In experiments, such high rates manifest themselves as precisely time-locked spikes with millisecond resolution (DeWeese et al., 2003; Wehr and Zador, 2003; Bolding and Franks, 2018; Gjoni et al., 2018). Recent studies suggest that cortical networks operate as inhibition-stabilized networks (ISNs) (Sanzeni et al., 2020; Sadeh and Clopath, 2021), in which the excitatory network is unstable in the absence of feedback inhibition (Tsodyks et al., 1997; Ozeki et al., 2009). To that end, we investigated how ensemble re-stabilization relates to the network operating regime at baseline and during stimulation. Whether a network is an ISN or not is mathematically determined by the real part of the leading eigenvalue of the Jacobian of the excitatory-to-excitatory subnetwork (Tsodyks et al., 1997). We computed the leading eigenvalue in our model incorporating STP and referred to it as ‘ISN index’ (Materials and methods; Appendix 2). We found that in networks with STP the ISN index can switch sign from negative to positive during external stimulation, indicating that the ensemble can transition from a non-ISN to an ISN (Figure 2F). Notably, this behavior is distinct from linear network models in which the network operating regime is independent of the input (Materials and methods). Whether this switch between non-ISN to ISN occurred, however, was parameter dependent and we also found network configurations that were already in the ISN regime at baseline and remained ISNs during stimulation (Figure 2—figure supplement 5). Thus, re-stabilization was largely unaffected by the network state and consistent with experimentally observed ISN states (Sanzeni et al., 2020). Theoretical studies have shown that one defining characteristic of ISNs in static excitatory and inhibitory networks is that injecting excitatory (inhibitory) current into inhibitory neurons decreases (increases) inhibitory firing rates, which is also known as the paradoxical effect (Tsodyks et al., 1997; Miller and Palmigiano, 2020). Yet, it is unclear whether in networks with STP, inhibitory stabilization implies paradoxical response and vice versa. We therefore analyzed the condition of being an ISN and the condition of having paradoxical response in networks with STP (Materials and methods; Appendix 2; Appendix 3). Interestingly, we found that in networks with E-to-E STD, the paradoxical effect implies inhibitory stabilization, whereas inhibitory stabilization does not necessarily imply paradoxical response (Figure 2G; Materials and methods), suggesting that having paradoxical effect is a sufficient but not necessary condition for being an ISN. In contrast, in networks with E-to-I STF, inhibitory stabilization and paradoxical effect imply each other (Appendix 2; Appendix 3). Therefore, paradoxical effect can be exploited as a proxy for inhibition stabilization for networks with STP we considered here. By injecting excitatory current into the inhibitory population, we found that the network did not exhibit the paradoxical effect before stimulation (Figure 2H, left; Figure 2—figure supplement 6). In contrast, injecting excitatory inputs into the inhibitory population during stimulation reduced their activity (Figure 2H, right; Figure 2—figure supplement 6). As demonstrated in our analysis, non-paradoxical response does not imply non-ISN (Figure 2—figure supplement 7; Materials and methods). We therefore examined the inhibition stabilization property of the ensemble by probing the ensemble behavior when a small transient perturbation to excitatory population activity is introduced while inhibition is frozen before stimulation and during stimulation. Before stimulation, the firing rate of the excitatory population slightly increases and then returns to its baseline after the transient perturbation (Figure 2—figure supplement 8). During stimulation, however, the transient perturbation leads to a transient explosion of the excitatory firing rate (Figure 2—figure supplement 8). These results further confirm that the ensemble shown in our example is initially a non-ISN before stimulation and can transition to an ISN with stimulation. By elevating the input level at the baseline in the model, the ensemble can be initially an ISN (Figure 2—figure supplement 5), resembling recent studies revealing that cortical circuits in the mouse V1 operate as ISNs in the absence of sensory stimulation (Sanzeni et al., 2020). Despite the fact that the supralinear input-output function of our framework captures some aspects of intracellular recordings (Priebe et al., 2004), it is unbounded and thus allows infinitely high firing rates. This is in contrast to neurobiology where firing rates are bounded due to neuronal refractory effects. While this assumption permitted us to analytically study the system and therefore to gain a deeper understanding of the underlying ensemble dynamics, we wondered whether our main conclusions were also valid when we limited the maximum firing rates. To that end, we carried out the same simulations while capping the firing rate at 300 Hz. In the absence of additional SFA or STP mechanisms, the firing rate saturation introduced a stable high-activity state in the ensemble dynamics which replaced the unstable dynamics in the uncapped model. As above, the ensemble entered this high-activity steady-state when stimulated with an external input above a critical threshold and exhibited persistent activity after stimulus removal (Figure 2—figure supplement 9). While weak SFA did not change this behavior, strong SFA resulted in oscillatory behavior during stimulation consistent with previous analytical work (Figure 2—figure supplement 9, van Vreeswijk and Hansel, 2001), but did not in stable steady-states commonly observed in biological circuits. In the presence of E-to-E STD or E-to-I STF, however, the ensemble exhibited transient evoked activity at stimulation onset that was comparable to the uncapped case. Importantly, the ensemble did not show persistent activity after the stimulation (Figure 2—figure supplement 9). Finally, we confirmed that all of these findings were qualitatively similar in a realistic spiking neural network model (Figure 2—figure supplement 10; Materials and methods). In summary, we found that neuronal ensembles can rapidly, nonlinearly, and transiently amplify inputs by briefly switching from stable to unstable dynamics before being re-stabilized through STP mechanisms. We call this mechanism nonlinear transient amplification (NTA) which, in contrast to balanced amplification (Murphy and Miller, 2009; Hennequin et al., 2012), arises from population dynamics with supralinear neuronal activation functions interacting with STP. While we acknowledge that there may be other nonlinear transient amplification mechanisms, in this article we restrict our analysis to the definition above. NTA is characterized by a large onset response, a subsequent ISN steady-state while the stimulus persists, and a return to a unique baseline activity state after the stimulus is removed. Thus, NTA is ideally suited to rapidly and nonlinearly amplify sensory inputs through recurrent excitation, like reported experimentally (Ko et al., 2011; Cossell et al., 2015), while avoiding persistent activity. Co-tuned inhibition broadens the parameter regime of NTA in the absence of persistent activity Up to now, we have focused on a single neuronal ensemble. However, to process information in the brain, several ensembles with different stimulus selectivity presumably coexist and interact in the same circuit. This coexistence creates potential problems. It can lead to multi-stable persistent attractor dynamics, which are not commonly observed and could have adverse effects on the processing of subsequent stimuli. One solution to this issue could be EI co-tuning, which arises in network models with plastic inhibitory synapses (Vogels et al., 2011) and has been observed experimentally in several sensory systems (Wehr and Zador, 2003; Froemke et al., 2007; Okun and Lampl, 2008; Rupprecht and Friedrich, 2018). To characterize the conditions under which neuronal ensembles nonlinearly amplify stimuli without persistent activity, we analyzed the case of two interacting ensembles. More specifically, we considered networks with two excitatory ensembles and distinguished between global and co-tuned inhibition (Figure 3A). In the case of global inhibition, one inhibitory population non-specifically inhibits both excitatory populations (Figure 3A, left). In contrast, in networks with co-tuned inhibition, each ensemble is formed by a dedicated pair of an excitatory and an inhibitory population which can have cross-over connections, for instance, due to overlapping ensembles (Figure 3A, right). Figure 3 with 1 supplement see all Download asset Open asset Co-tuned inhibition broadens the parameter regime of NTA in the absence of persistent activity. (A) Schematic of two neuronal ensembles with global inhibition (left) and with co-tuned inhibition (ri