Dominant role of adult neurogenesis‐induced structural heterogeneities in driving plasticity heterogeneity in dentate gyrus granule cells

Abstract Neurons and synapses manifest pronounced variability in the amount of plasticity induced by identical activity patterns. The mechanisms underlying such plasticity heterogeneity, which have been implicated in context‐specific resource allocation during encoding, have remained unexplored. Here, we employed a systematic physiologically constrained parametric search to identify the cellular mechanisms behind plasticity heterogeneity in dentate gyrus granule cells. We used heterogeneous model populations to ensure that our conclusions were not biased by parametric choices in a single hand‐tuned model. We found that each of intrinsic, synaptic, and structural heterogeneities independently yielded heterogeneities in synaptic plasticity profiles obtained with two different induction protocols. However, among the disparate forms of neural‐circuit heterogeneities, our analyses demonstrated the dominance of neurogenesis‐induced structural heterogeneities in driving plasticity heterogeneity in granule cells. We found that strong relationships between neuronal intrinsic excitability and plasticity emerged only when adult neurogenesis‐induced heterogeneities in neural structure were accounted for. Importantly, our analyses showed that it was not imperative that the manifestation of neural‐circuit heterogeneities must translate to heterogeneities in plasticity profiles. Specifically, despite the expression of heterogeneities in structural, synaptic, and intrinsic neuronal properties, similar plasticity profiles were attainable across all models through synergistic interactions among these heterogeneities. We assessed the parametric combinations required for the manifestation of such degeneracy in the expression of plasticity profiles. We found that immature cells showed physiological plasticity profiles despite receiving afferent inputs with weak synaptic strengths. Thus, the high intrinsic excitability of immature granule cells was sufficient to counterbalance their low excitatory drive in the expression of plasticity profile degeneracy. Together, our analyses demonstrate that disparate forms of neural‐circuit heterogeneities could mechanistically drive plasticity heterogeneity, but also caution against treating neural‐circuit heterogeneities as proxies for plasticity heterogeneity. Our study emphasizes the need for quantitatively characterizing the relationship between neural‐circuit and plasticity heterogeneities across brain regions.


| INTRODUCTION
Neurons and synapses of the same subtype receiving identical plasticity-inducing activity patterns do not manifest identical levels of plasticity. Instead, they exhibit plasticity heterogeneity across synapses and neurons, manifesting as pronounced variability in the observed changes. There are several lines of evidence from in vitro and in vivo electrophysiological experiments for such plasticity heterogeneity, spanning different neuronal and synaptic subtypes (Beck et al., 2000;Bliss & Lomo, 1973;Davis et al., 2004;Greenstein et al., 1988;Kobayashi et al., 2013;Koranda et al., 2008;Larson & Munkacsy, 2015;Li et al., 2017;McHugh et al., 2007;Pavlides et al., 1988;Rathour & Narayanan, 2019;Shors & Dryver, 1994;Sjostrom et al., 2008;Wang et al., 1997). Although such plasticity heterogeneity has typically been overlooked in analyzing the impact of plasticity protocols, a growing body of experimental evidence identifies crucial roles for plasticity heterogeneity in neural encoding and storage. Specifically, the ability of neurons and synapses to undergo differential plasticity is critical for context-specific recruitment/allocation of a subset of neurons and synapses during encoding processes (Aimone et al., 2014;Dieni et al., 2013;Ge et al., 2007;Huckleberry & Shansky, 2021;Josselyn & Frankland, 2018;Josselyn & Tonegawa, 2020;Lau et al., 2020;Lodge & Bischofberger, 2019;Park et al., 2016;Pignatelli et al., 2019;Schmidt-Hieber et al., 2004;Yiu et al., 2014). The lack of plasticity heterogeneity would result in a scenario where all neurons and synapses undergo similar amount of plasticity for any given context. Such a scenario would erase the possibility of sparse and context-specific recruitment of neural resources. Despite these well-recognized roles of plasticity heterogeneity in context-specific resource allocation, the mechanisms underlying these heterogeneities have not been assessed. Furthermore, there are postulates and lines of evidence for heterogeneities in intrinsic excitability playing a role in determining selective resource allocation (Aimone et al., 2014;Dieni et al., 2013;Ge et al., 2007;Huckleberry & Shansky, 2021;Josselyn & Frankland, 2018;Josselyn & Tonegawa, 2020;Lau et al., 2020;Lodge & Bischofberger, 2019;Park et al., 2016;Pignatelli et al., 2019;Schmidt-Hieber et al., 2004;Yiu et al., 2014). However, the quantitative link between such cellularscale heterogeneities and plasticity heterogeneity has not been systematically assessed.
Together, GCs provided an efficient substrate for assessing the impact of well-characterized biophysical and structural heterogeneities on the emergence of plasticity heterogeneity.
In this study, we systematically explored the cellular-scale origins of heterogeneities in the synaptic plasticity profiles of DG GCs through an unbiased exploration of heterogeneities in their intrinsic, synaptic, and structural properties. We ensured that our analyses associated with each of these heterogeneities were constrained by characteristic physiological properties of mature and immature GCs.
We assessed the impact of these forms of heterogeneities on plasticity profiles obtained with two well-established protocols for inducing synaptic plasticity in DG GCs: the 900-pulses protocol spanning a range of induction frequencies (Kobayashi et al., 2013;Koranda et al., 2008;Wang et al., 1997), and the theta-burst stimulation protocol (Beck et al., 2000;Davis et al., 2004;Greenstein et al., 1988;Larson & Munkacsy, 2015;McHugh et al., 2007;Pavlides et al., 1988;Shors & Dryver, 1994). We found that each form of intrinsic, synaptic, and structural heterogeneity independently resulted in plasticity heterogeneities, with either protocol for plasticity induction. Importantly, when immature and mature neuron populations were individually analyzed, we found that heterogeneities in intrinsic excitability were insufficient to impose strong constraints on plasticity-related measurements. However, when the entire population covering mature and immature cells were analyzed together, there were strong relationships between intrinsic excitability and measurements associated with synaptic plasticity.
We show that the expression of heterogeneities in all of structural, synaptic, and intrinsic neuronal properties does not necessarily have to translate to heterogeneities in synaptic plasticity profiles.
Specifically, we demonstrate that very similar plasticity profiles could be achieved with disparate combinations of neuronal passive properties, ion-channel properties, calcium-handling mechanisms, synaptic strength, and neural structure of DG GCs of different ages. When observed independently, these properties manifested widespread heterogeneities with weak pairwise relationships. However, when seen together, these heterogeneities synergistically interacted with each other to achieve the functional goal of degeneracy in synaptic plasticity profiles. These analyses extend degeneracy in DG GCs to the concomitant emergence of plasticity profiles and of several neural intrinsic properties. Importantly, this form of degeneracy encompasses cellular-scale intrinsic, synaptic, and structural heterogeneities spanning different age groups of GCs in a physiologically constrained manner. These analyses also showed that synaptic plasticity in the useful physiological range could be achieved in immature cells even with the weak synaptic strengths that they are known to express, owing to strong relationships with intrinsic excitability measurements.
Together, our analyses demonstrate that intrinsic, synaptic, and structural heterogeneities could either individually or through synergistic interactions among them, drive plasticity heterogeneity in DG GCs. Importantly, our analyses demonstrate that similar plasticity profiles could be achieved despite the concomitant expression of all forms of neural-circuit heterogeneities. These observations caution against treating the manifestation of neural-circuit heterogeneities as direct evidence for the expression of plasticity heterogeneities. Our results also highlighted the dominance of structural heterogeneities, introduced by adult neurogenesis, in introducing plasticity heterogeneity that is essential for context-specific resource allocation in the DG. From a broader perspective, our analyses call for systematic characterization and analyses of plasticity heterogeneities across different brain regions. Such analyses should probe the mechanistic origins of plasticity heterogeneities and assess their implications for contextspecific neural coding of learned behavior and memory storage.

| MATERIALS AND METHODS
Granule cells in the DG exhibit heterogeneities in neuronal properties (intrinsic heterogeneity), in synaptic connections (synaptic heterogeneity), and structural properties including dendritic arborization and surface area (structural heterogeneity). In this study, our goal is to explore the impact of these heterogeneities on synaptic plasticity profiles, employing conductance-based models for DG GCs. Assessment of plasticity profiles involve long-term simulations and the complexities associated with incorporating different forms of heterogeneities in a population of conductance-based models (as opposed to a single model with fixed structure and fixed synaptic strengths) implied large computational costs. Thus, we employed single-compartmental conductance-based models to assess the impact of different forms of biophysical and structural heterogeneities on synaptic plasticity induced through two extensively employed plasticity-induction protocols.
2.1 | Heterogeneities in intrinsic properties of a physiologically constrained granule cell model population Granule cells in the DG manifest pronounced heterogeneities in their intrinsic properties (Aradi & Holmes, 1999;Krueppel et al., 2011;Lubke et al., 1998;Mishra & Narayanan, 2020;Santhakumar et al., 2005). The physiologically constrained conductance-based heterogeneous population of granule cell model was obtained from an earlier study (Mishra & Narayanan, 2019). The details of building this population of models that manifested characteristic electrophysiological properties of GCs, employing a multiparametric multiobjective stochastic search (MPMOSS) algorithm are identical to the previous study (Mishra & Narayanan, 2019). Briefly, the dimensions of single cylindrical base model were set to 63 μm diameter (diam) and 63 μm length (len) ( Figure   1a). The resting membrane potential of model cell was set to À75 mV, with specific membrane resistance (R m ) of 38 kΩ.cm 2 and specific membrane capacitance (C m ) of 1 μF.cm À2 . The dimensions of the cylindrical compartment were set toward achieving a passive input resistance of 305 MΩ (R m /(π Â diam Â len) = 38 Â 10 3 Â 10 À2 Â 10 À2 /(π Â 63 Â 10 À6 Â 63 Â 10 À6 ) = 305 MΩ), matching the experimental value of 309 ± 14 MΩ obtained with pharmacological blockers of HCN channels (Chen, 2004). This passive input resistance was consequent to the leak conductance (specified as R m ) and the surface area of the compartment, and will be validated against the electrophysiological values of active input resistance (i.e., in the presence of subthreshold ion channels).
These passive parameters also resulted in a charging time constant (R m C m ) of 38 ms (Schmidt-Hieber et al., 2007).
The GC model is comprised of nine different regenerative and restorative conductances: fast sodium (NaF), hyperpolarizationactivated cyclic-nucleotide-gated (HCN), L-type calcium (CaL), N-type calcium (CaN), T-type calcium (CaT), delayed rectifier potassium (KDR), A-type potassium (KA), big conductance (BK), and small conductance (SK) calcium activated potassium. Hodgkin-Huxley (HH) or Goldman-Hodgkin-Katz (GHK) formulations (Goldman, 1943;Hodgkin & Huxley, 1952;Hodgkin & Katz, 1949) were employed to model these voltage-and/or calcium-gated ion channels (Mishra & Narayanan, 2019). The GHK formulation was used to model calcium conductances, with intracellular and extracellular calcium concentration set at 50 nM and 2 mM, respectively. The reversal potential values for Na, K, and HCN channels were set as +55, À90, and À30 mV, respectively. Cytosolic calcium concentration and its evolution with time was dependent on calcium current and its decay, and the mechanism was adopted from the formulation (Carnevale & Hines, 2006;Destexhe et al., 1993;Narayanan & Johnston, 2010;Poirazi et al., 2003): The multiple parameters and their ranges for the stochastic search employed for finding the 126 valid granule cells (Mishra & Narayanan, 2019 where F is the Faraday's constant, the calcium decay constant in GCs was given by τ Ca with a default value of 160 ms, dpt represented the depth of the shell into which calcium influx occurred and was taken as 0.1 μm, and Ca ½ ∞ = 50 nM was considered as the steady-state value of Ca ½ c . We generated 20,000 models of GC through a stochastic search from a parametric space comprised of 40 different parameters (Table   1): 38 parameters associated with nine active conductances along with 2 passive neuronal parameters. The GC models were declared valid once they fall within the range of nine physiologically constrained measurements (Table 2): input resistance (R in ), sag ratio, firing rate at 50 pA (f 50 ) and 150 pA (f 150 ) current injection, spike frequency adaptation, action potential (AP) amplitude, AP threshold, AP half width, and fast afterhyperpolarization. The validation process resulted in 126 valid models (126/20,000, implying a 6.3% population of valid models) that manifested characteristic electrophysiological properties of GCs but exhibited pronounced heterogeneities in channel composition and other biophysical parameters (Mishra & Narayanan, 2019).
This constitutes an instance of ion-channel degeneracy (Goaillard & Marder, 2021;Mishra & Narayanan, 2019, 2021aRathour & Narayanan, 2019) in the emergence of cellular-scale properties and provided 126 GC models that were endowed with signature heterogeneities in their intrinsic properties. In our analyses, this population of 126 GC models is identical to the models from Mishra and Narayanan (2019) and was employed as the substrate for assessing the impact of intrinsic heterogeneities on synaptic plasticity profiles.

| Properties and associated heterogeneities in synapses impinging on granule cell models
We modeled a canonical synapse impinging on the postsynaptic GC neuron as two co-localized excitatory synaptic receptors: α-amino- Note: The first nine measurements were employed to validate the 126 (of the 20,000 samples) intrinsically heterogeneous model neurons (Mishra & Narayanan, 2019), whereas the last two measurements were validated for the 126 models ( Figure 1) with electrophysiological bounds derived from Mishra and Narayanan (2020). These 126 models showed characteristic electrophysiological properties and neuron-to-neuron heterogeneity that were comparable with electrophysiological recordings (Mishra & Narayanan, 2019, 2021a. These 126 valid models were sufficient to demonstrate that disparate combinations of ion channels could yield very similar characteristic properties (Mishra & Narayanan, 2019, 2021b. Importantly, the parametric values of these 126 models spanned the entire valid range of each parameter suggesting the absence of any parametric clustering (Mishra & Narayanan, 2019), together demonstrating the expression of ion-channel degeneracy (Mishra & Narayanan, 2021a).
3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptor (AMPAR) and N-methyl-D-aspartate (NMDA) receptor (NMDAR) with an NMDA: AMPA ratio value of 1.5. The current through AMPAR and NMDAR as a function of voltage and time are modeled using the GHK formulation (Goldman, 1943;Hodgkin & Katz, 1949) as a sum of current generated by sodium and potassium ions (Anirudhan & Narayanan, 2015;Honnuraiah & Narayanan, 2013;Narayanan & Johnston, 2010): where Here, P AMPAR represents the maximum permeability of the receptor, also used as a synaptic parameter to incorporate synaptic heterogeneity. w represents the synaptic weight parameter that would be updated and monitored as a function of time to quantify positive and negative weight changes based on the plasticity protocol (see below).
The default value of initial weight, w init was set to 0.25. The sodium (P Na ) and potassium (P K ) permeability values were set to be equal (P Na : P K = 1:1) based on experimental observations. The default values for intracellular and extracellular concentration (mM) of specific ions were Na ½ i = 18, Na ½ o = 140, K ½ i = 140, K ½ o = 5, which led to equilibrium potential of +55 mV and À90 mV for Na and K, respectively. s t ð Þ guides the kinetics of AMPA current as represented using the twoexponential formulation: where a represents normalization constant so that 0 < s(t) < 1. τ r and τ d denote the rise and decay time constants associated with AMPA receptor with values of 2 and 10 ms, respectively. Synaptic heterogeneities were introduced into the population of models by altering the permeability value of P AMPAR .
The current through NMDA receptor depended on sodium, potassium, and calcium ions and was modeled as follows using the GHK formulation: where P NMDAR denotes the maximum permeability of the NMDA receptor and was defined as the product of P AMPAR , w init , and the value of NMDA:AMPA ratio. The permeability ratios of three ions for NMDAR are set as P Ca : P Na : P K = 10.6:1:1 (Canavier, 1999;Mayer & Westbrook, 1987). The s t ð Þ function was same as for AMPAR with τ r = 5 ms and τ d = 50 ms. The concentration values in mM are Na dependence of NMDAR currents on extracellular magnesium concentration ( Mg ½ o ) and voltage (Jahr & Stevens, 1990). The current through NMDAR did not undergo plasticity. counterparts (Mishra & Narayanan, 2019). Electrophysiologically, R in of mature and immature cells have been measured to be in the $100-300 MΩ and $3-6 GΩ ranges, respectively (Heigele et al., 2016;Mishra & Narayanan, 2020, 2021aOverstreet-Wadiche, Bromberg, et al., 2006;Pedroni et al., 2014;Schmidt-Hieber et al., 2004). Reducing the diameter of the models in neural population increased neuronal excitability, reflecting as increased R in and increased firing rate. To assess the impact of structural heterogeneities on synaptic plasticity profiles, we varied the diameter of the 126 neurons in the model population from 1 to 63 μm. A diameter range of 2-9 μm was chosen because this yielded R in values that matched the experimental 3-6 GΩ range for immature neurons and was considered representative of the immature neuronal models (Mishra & Narayanan, 2019).

| Intrinsic measurements
The 126 GC models were selected based on the nine physiological measurements employed to characterize the valid GC population (Table 2; Mishra & Narayanan, 2019). In addition to these, we introduced two more sub-threshold measurements (impedance amplitude and temporal summation ratio) to test the robustness of these intrinsically heterogeneous models (Figure 1c,d) and to compare their role in regulating plasticity profiles. Specifically, we employed input resistance (R in ), firing frequency to pulse current injections, sag ratio, impedance amplitude, and temporal summation as intrinsic measurements towards relating them to plasticity profiles. R in was measured as the slope of a linear fit to the I-V plot. The I-V plot was obtained by plotting the steady state value of voltage response as a function of 11 different current pulses where the amplitude varied from À50 to +50 pA in steps of 10 pA ( Figure   1b). As GC models with lower diameters manifested high excitability, R in was computed in response to hyperpolarizing current pulses ranging from À50 to 0 pA in steps of 10 pA, to avoid spike generation. To characterize the impedance amplitude profiles of these F I G U R E 1 Model components of dentate gyrus granule cells and illustration of intrinsic heterogeneities across different physiological measurements. (a) Conductance-based single compartmental model of granule cell expressing different inward and outward voltage-dependent ion-channel currents, receiving excitatory inputs modeled as α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-Daspartate (NMDA) receptor currents. (b-f) Different intrinsic physiological measurements are employed to define the valid population of granule cells (GC) models (N GC = 126). (b) Left, voltage traces in response to current pulses of amplitude À50 pA to +50 pA, in steps of 10 pA. Right, input resistance (R in ), calculated as the slope of the V-I curve obtained by plotting the steady-state voltage responses against injected current amplitudes. (c) Top, a chirp current stimulus of 50 pA peak-to-peak amplitude with linearly increasing frequency from 0 to 15 Hz in 15 s depicted along with the respective voltage response. Bottom, the impedance amplitude profile obtained from the chirp current and voltage response shown above. (d) Left, voltage response of a GC model to current input comprised five α-EPSCs arriving at 20 Hz, to compute temporal summation ratio (S α ). S α is the ratio of the voltage amplitude in response to the fifth α-EPSC to that of the first α-EPSC. (e) Left, membrane potential in response to 50 pA hyperpolarizing current pulse to calculate sag ratio. Sag ratio is the ratio between the steady-state voltage response and the peak voltage response. (f) Left, firing pattern and firing rate in response to the 150 pA depolarizing current pulse of 1 s duration. Across all panels in (b-f), the right panels show beeswarm plots depicting heterogeneities in the respective measurement across all 126 models. The heterogeneous population of 126 GC models employed here is from Mishra and Narayanan (2019), with additional characterization involving new intrinsic measurements added to the validation process. models, we injected chirp stimulus, a frequency-dependent current input with linearly increasing frequency from 0 to 15 Hz in 15 s of constant amplitude (Mishra & Narayanan, 2020). The impedance profile Z f ð Þ was computed as the ratio of the Fourier transform of voltage response to the Fourier transform of chirp current as a function of frequency ( Figure 1c). The impedance amplitude profile was calculated as follows: where Þrefer to the real and imaginary parts of the impedance Z f ð Þ, respectively, as functions of the frequency f. The maximum value of impedance across all frequencies was measured as the maximum impedance amplitude (jZj max ).
Temporal summation ratio (S α ) was computed by injecting cur-

| Synaptic plasticity protocols and weight evolution
The synaptic weight parameter w governing current through AMPAR depended on the intracellular calcium concentration as follows, based on the calcium control hypothesis (Shouval et al., 2002): where η Ca ½ i À Á represents learning rate dependent on calcium concentration, which is inversely related to learning time constant τ Ca ½ i À Á as follows: τ Ca where P 1 = 1 s, P 2 = 0.1 s, P 3 = P 2 Â 10 À4 , and P 4 = 3. These values when substituted in Equation (12) sets the learning time constant to , the function that governed the calcium-dependent weight update mechanism, was defined as (Shouval et al., 2002): where α 1 = 0.35, α 2 = 0.55, β 1 = β 2 = 80. For all the weight update equations, Ca ½ i were set as the deflection from the resting value of Ca ½ i .
Using this framework, we analyzed the direction and strength of plasticity in w using two well-established synaptic plasticity protocols in DG neurons: the 900-pulses protocol with varying induction frequencies (Kobayashi et al., 2013;Koranda et al., 2008;Wang et al., 1997) and the theta burst stimulation (TBS) protocol (Beck et al., 2000;Davis et al., 2004;Greenstein et al., 1988;Larson & Munkacsy, 2015;McHugh et al., 2007;Pavlides et al., 1988;Shors & Dryver, 1994). The 900-pulses pro- For TBS, the synapse was stimulated with a burst of five action potentials at 100 Hz, and this burst was repeated 150 times at 200 ms interval (theta frequency) each ( Figure 5a). This was done to achieve steady-state values for Ca ½ i and w (Ashhad & Narayanan, 2013). The percentage change in w at the end of this protocol in comparison to its initial value (w init ¼ 0:25) was employed to quantify plasticity induced with TBS. For both plasticity induction protocols, we employed a spike train generator as an input source to mimic presynaptic activity.
These synaptic plasticity protocols and the rules for updating synapses were chosen from the perspective of their relevance to synapses in the DG GCs. Specifically, the two protocols employed here are well-established routes to induce synaptic plasticity in DG GCs (Beck et al., 2000;Davis et al., 2004;Greenstein et al., 1988;Kobayashi et al., 2013;Koranda et al., 2008;Larson & Munkacsy, 2015;McHugh et al., 2007;Pavlides et al., 1988;Shors & Dryver, 1994;Wang et al., 1997). The calcium-dependent plasticity rule employed here is a BCM-like plasticity rule that has been F I G U R E 2 Intrinsic heterogeneities in the granule cell population translates to heterogeneities in their BCM-like synaptic plasticity profiles, when synaptic properties were fixed across models. (a) Plot of the Ω-function based on the calcium control hypothesis that regulates level of plasticity as a function of intracellular Ca 2+ concentration (Equation 11). (b) Evolution of synaptic weight as a function of time, obtained by employing 900-pulses protocol of different induction frequencies in a granule cell (GC) model. Note that all plots initialize at w init ¼ 0:25 and evolve to reach their respective steady-state value. The duration of each plot spans 900 pulses at the specified induction frequency f i . (c) BCMlike synaptic plasticity profile obtained by plotting the percentage change in synaptic weight parameter after stimulation with 900-pulses of different induction frequencies ranging from 0.5 to 25 Hz. The color-coded points correspond to the different induction frequencies shown in panel b. Arrows point to θ m , Δw 1 and Δw 10 . Δw 1 and Δw 10 represent the change in synaptic weight value for induction frequencies of 1 and 10 Hz, respectively; θ m , the modification threshold, is the induction frequency at which the plasticity profile switches from inducing LTD to inducing LTP. (d-e) Same as (c), for all the 126 GC models for two different values of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR) permeability: 1000 nm/s (d) and 1400 nm/s (e). (f) Beeswarm plots of modification threshold for all GC models, for different values of AMPAR permeability. Note that with specific values of AMPAR permeability, there were models that did not manifest a θ m in the tested range of frequencies, thus resulting in lesser number of models for each AMPAR permeability values (N = 100, 121, 121, 120, 112, 94, 63, 39 left to right).
The rationale behind our choice of the calcium-control hypothesis is the match between the plasticity profile obtained with the 900-pulses protocol in DG GCs (Kobayashi et al., 2013;Koranda et al., 2008;Wang et al., 1997) and the calcium-dependent plasticity profile explained by the BCM rule (Bienenstock et al., 1982;Cooper & Bear, 2012;Shouval et al., 2002).

| Computer simulations and analysis
We employed NEURON as the simulation environment (Carnevale & Hines, 2006) for executing all the simulation at V RMP (À75 mV) with fixed temperature set at 34 C. We used the integration were adopted from the definitions provided in the study by Evans (1996).

| RESULTS
We employed a physiologically realistic conductance-based population of GC models (N GC =126), endowed with intrinsic heterogeneities and expressing ion-channel degeneracy at the cellular-scale (Mishra & Narayanan, 2019), to assess the impact of neural heterogeneities on synaptic plasticity profiles. In this population, we introduced synaptic heterogeneities by altering afferent synaptic strength, and structural heterogeneities by changing the surface area of the model population.
We employed two well-established synaptic plasticity protocols, namely the BCM-like 900-pulses protocol with different induction frequencies and the TBS protocol, to examine the impact of these three forms of neural heterogeneities in the regulation of synaptic plasticity rules in DG GCs. We present results obtained through systematic incorporation of these different forms of heterogeneities, both independently and synergistically, into a physiologically validated GC model population.

| GC models showed robustness for nonvalidated measurements and manifested heterogeneities in intrinsic measurements
The 126 GC models employed in this study were derived from an unbiased stochastic search spanning 40 parameters (Table 1), sampling 20,000 randomized models (Mishra & Narayanan, 2019). Of the 20,000 models, these 126 models were previously validated based on nine different characteristic electrophysiological signatures (Table 2) of DG GCs (Mishra & Narayanan, 2019). Prominent among these measurements are input resistance (R in , range 140-225 MΩ; Figure 1b), sag ratio (range 0.9-1; Figure 1e) and firing rate at 150 pA (range 10-15 Hz; Figure 1f), which manifested heterogeneities. In addition to these, here we characterized two more experimentally obtained subthreshold measurements of excitability to assess their relationship to the induction of synaptic plasticity: impedance amplitude and temporal summation ratio (Figure 1c,d). Whereas temporal summation of postsynaptic potentials constitutes an important measurement that governs calcium influx and thereby synaptic plasticity (Narayanan & Johnston, 2010;Nolan et al., 2004), impedance is a measure of excitability for time-varying signals (Narayanan & Johnston, 2008).
Although the 126 GC models were initially not validated against these two measurements, here we found that these measurements in the models were within the range of their electrophysiological counterparts (Mishra & Narayanan, 2020). To address these questions, we first executed an algorithm, independently for each of the 126 models, that identified the value of baseline synaptic strength (P AMPAR ) that yielded a synaptic plasticity profile with the modification threshold around 10 Hz (9:75 ≤ θ m ≤ 10:25).
Despite the considerable heterogeneities in intrinsic properties, we found that altering P AMPAR was sufficient to achieve similar synaptic plasticity profiles across all 126 models (Figure 4a) with θ m falling within the tight bound (Figure 4b). The considerable heterogeneities in intrinsic properties, however, manifested as heterogeneity in the P AMPAR value required to achieve similar plasticity profiles. The value of P AMPAR required to achieve similar plasticity profiles (referred to as threshold P AMPAR ) spanned a wide range (Figure 4c), with the heterogeneity almost spanning an order of magnitude across models (450-3100 nm/s). Thus, although changes in P AMPAR resulted in changes to the plasticity profile across models (Figure 2f), specific co-expression of heterogeneities in synaptic ( Figure 4c) and intrinsic (Figure 1) properties could result in similar plasticity profiles (Figure 4a,b).
Did the emergence of similar plasticity profiles require strong constraints on the relationship between synaptic strength and intrinsic excitability of the models? Were there strong relationships between synaptic strength and any of the biophysical parameters that defined the models that yielded similar synaptic plasticity profiles? To address these, we first computed pairwise correlation coefficients between the P AMPAR value that was required to obtain similar plasticity profiles (from Figure 4c) and five intrinsic measurements of the respective models (from Figure 1) and found them to be weakly correlated ( Figure 4d; À0.03 < R < 0.02). We next plotted pair wise scatter plot matrix between these P AMPAR values (from Figure 4c) and the 40 different intrinsic parameters that defined these 126 models to explore possible parametric dependencies (Figure 4e). We found these pairwise correlation coefficients to be weak (À0.5 < R < 0.5; Figure   4f) Hz) to induce synaptic plasticity in these models. Specifically, the impact of plasticity induction was assessed in the 126 intrinsically heterogeneous models, with the diameter changes spanning 3-65 μm, which incorporated an additional layer of structural heterogeneity into each of these models. A third layer of synaptic heterogeneity was introduced by varying the baseline AMPAR permeability P AMPAR value, together providing us an experimental design that allowed us to assess the impact of all the three prominent neural-circuit heterogeneities on the synaptic plasticity profile (Figure 7a-e).
Considering an example of a single granule cell model, we found that altering the diameter of the neuron had a dramatic impact on the synaptic plasticity profile even when P AMPAR was set at a fixed value Modification threshold (θ m ), percentage weight change at 1 Hz (Δw 1 ) and 10 Hz (Δw 10 ) and measurements of intrinsic excitability: R in , f 100 , and f 150 for two AMPAR permeability values: P AMPAR = 400 nm/s (f) and 600 nm/s (g), across six diameters values (3, 9, 30, 40, 50, and 63 μm). The scatter plots are overlaid to corresponding color-coded pairwise correlation coefficients representing weak pairwise correlations across diameters and permeability values. Note that θ m did not fall within the tested range of induction frequencies for different models with different diameter values, thus resulting in lesser points for certain diameter values. The axes ranges for each measurement span the entire range of the respective measurements and are different across different plots. Figure 7a). For a fixed value of P AMPAR , we found that the modification threshold increased as a function of diameter, albeit manifesting considerable heterogeneity in the modification threshold for a given diameter value across different models (Figure 7e). For several models with diameters of 10, 40, 50, and 63 μm, the modification threshold (with P AMPAR = 475 nm/s) was not within the tested range of induction frequencies (0.5-25 Hz), thus resulting in lesser number of models for those diameters (Figure 7e).
Were there strong relationships between intrinsic and synaptic plasticity measurements across these models across different diameters and different values of P AMPAR ? To answer this, we employed three intrinsic measurements (R in , f 100 and f 150 ) and three measurements of synaptic plasticity (θ m , Δw 1 , and Δw 10 ), each measured for six diameter values (3,9,30,40,50, and 63 μm) and two P AMPAR values (Figure 7f,g). We computed Pearson's correlation coefficients between the intrinsic and synaptic plasticity measurements and found weak pair wise correlations between intrinsic and plasticity measure- To assess these questions, we first selected six intrinsically distinct GC models (from the population of 126 models) and assigned different values of diameters to each of these six models. We then employed an algorithm to find a synaptic permeability value (P AMPAR ) that yielded plasticity profiles endowed with their modification threshold at $10 Hz with the 900-pulse protocol (Figure 8a). We found the synaptic plasticity profiles for each of these six models, endowed with their respective P AMPAR provided by the algorithm, to be similar across the entire range of induction frequencies (0.5-25 Hz) (Figure 8a). We then plotted each of the 42 parameters underlying these six models (40 intrinsic parameters in Table 1, diameter as the structural parameter, and P AMPAR governing the synapse) and found each of them to span their respective ranges (Figure 8b). These analyses illustrate that models built with very different structural, intrinsic, and synaptic properties (Figure 8b) could together yield very similar synaptic plasticity profile (Figure 8a), thus demonstrating the emergence of plasticity degeneracy despite widespread variability in all underlying parameters.
We expanded the scope of our analyses to span all 126 intrinsically heterogeneous models, each spanning six diameter values (3,9,30,40,50, and 63 μm) and employed our algorithm to find a P AMPAR that would yield a modification threshold of $10 Hz (9:75 ≤ θ m ≤ 10:25) in each of these (126 Â 6 ¼ 756) models ( Figure 9a). We were able to find P AMPAR values that yielded similar modification thresholds, with the required P AMPAR increasing with increase in diameter (Figure 9b). We did not find strong correlations between the P AMPAR value required for achieving similar plasticity profiles and the respective intrinsic measurements

| Synergistic interactions between synaptic, intrinsic, and structural heterogeneities governed TBS-induced synaptic plasticity
We repeated our analyses on the impact of the three forms of heterogeneities with the TBS protocol. First, we found that heterogeneities in structural properties could alter the amount of synaptic plasticity achieved with TBS across the intrinsically heterogeneous model population, when structural heterogeneity was introduced by altering diameters to six different values representative of immature and mature granule cell populations. For these analyses, we fixed the P AMPAR value and found that the amount of plasticity obtained reduced with increasing value of diameter (Figure 10a, left), thus demonstrating a lower threshold on P AMPAR for inducing LTP in immature neurons. We also found that there was no correlation between the amount of plasticity achieved and the respective intrinsic properties, for each value of diameter assessed (Figure 10a, right). Second, to explore plasticity degeneracy with the TBS protocol, we next found P AMPAR values that yielded similar levels of synaptic plasticity of $150% (148%-152%) for each of the 126 intrinsically heterogeneous models, with six different values of diameters (Figure 10b,c). The P AMPAR value required for achieving similar plasticity increased with increase in diameter and did not manifest strong correlations with either the respective intrinsic measurements (Figure 10d) or the intrinsic parameters (Figure 10e) for each value of the diameter.
Together, our analyses demonstrated that each of intrinsic ( Figures   2d-f, 5d, 7e, and 10a), synaptic (Figures 2d-f, 5c,d, and 7c,d), and structural (Figures 7e and 10a) heterogeneities could independently introduce heterogeneities in the plasticity profiles, irrespective of the protocol employed. However, when they coexpress, these disparate forms of heterogeneities could synergistically interact with each other to yield very similar plasticity profiles (Figures 4a-c, 6a,b, 8, 9a,b and 10b,c), irrespective of the induction protocol employed. Across our analyses spanning different plasticity protocols, assessing heterogeneities or degeneracy in plasticity profiles, we did not find strong correlations between synaptic properties plotted against intrinsic measurements (Figures 3, 4d, 5e, 6c, 7f,g, 9c, and 10d) or intrinsic parameters ( Figures   4e, 6d, 9d, and 10e) of the model populations. These results suggested that the measurements of intrinsic excitability and temporal summation are not sufficiently strong to impose specific synaptic plasticity profiles.
3.9 | Importance of adult neurogenesis-induced structural heterogeneities in lowering plasticity induction threshold and recruiting engram cells based on intrinsic excitability In our analyses thus far, we noted that intrinsic excitability parameters were not strong enough to constrain synaptic plasticity induction, F I G U R E 8 Illustration of degeneracy in the emergence of plasticity profiles spanning biophysical, structural, and synaptic parameters using six models. (a) Frequency-dependent plasticity profiles plotted for six intrinsically disparate models with different diameters and P AMPAR values yield similar plasticity profile with modification threshold at $10 Hz. (b) Plots, for each of the six models shown in panel a, of the 40 intrinsic passive and active properties (listed in Table 1 with units), the diameter (in μm) and the P AMPAR (in nm/s) values required to get the modification threshold to be $10 Hz. The plots for each of the 40 intrinsic parameters (Table 1) and diameter (1-63 μm) span their entire search range. Note that the ranges of each parameter across the six models are highly variable (b), spanning a large portion of the parameter's search range, despite the similarity of the plasticity profiles (a) with a consistent lack of strong correlations between synaptic plasticity measurements and intrinsic excitability (Figures 4-7, 9, and 10).  (Figure 11a-d). Specifically, our analyses showed that the amount of induced plasticity was higher in neurons with high excitability (Figure 11a,b) and that the plasticity F I G U R E 9 Emergence of plasticity degeneracy due to synergistic interactions between age-dependent structural, synaptic, and intrinsic heterogeneities with weak pairwise correlations. (a) Plot representing the distribution of modification threshold for all GC models across different diameters to obtain modification threshold of $10 Hz by adjusting α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR) permeability for each model. There are lines of evidence that the synaptic strength of inputs to immature DG GCs is lower compared to their mature counterparts F I G U R E 1 0 Heterogeneities and degeneracy in synaptic plasticity achieved with theta burst stimulation (TBS) protocol in models endowed with age-dependent structural, synaptic, and intrinsic heterogeneities. (a) Left, age-dependent structural heterogeneity in the population of GC models translated to heterogeneity in the amount of plasticity achieved with tTBS protocol when baseline synaptic strength was fixed to 81 nm/ s. Shown is the amount of plasticity achieved for models in the intrinsically heterogeneous model population, with the diameter altered to assess the impact of structural heterogeneities. Right, pairwise scatter plots between different plasticity measurements associated with TBS versus measurements of intrinsic excitability: R in , f 100 , and f 150 for a fixed value of baseline synaptic strength, P AMPAR = 81 nm/s, across six diameter values (3, 9, 30, 40, 50, and 63 μm). The scatter plots are overlaid to corresponding color-coded pairwise correlation coefficients representing weak pairwise correlations across diameters and permeability values. The axes ranges for each measurement span the entire range of the respective measurements and are different across different plots. F I G U R E 1 1 The dominant role of structural heterogeneities in regulating plasticity profiles with the BCM-like and theta burst stimulation (TBS) plasticity protocols. (a,b) Percentage weight change at 1 Hz (Δw 1 ) with the 900-pulses protocol plotted against input resistance (a) and firing rate for 100 pA current injection (b) for models in the intrinsically heterogeneous population, with each model assessed at six different diameter values (3, 9, 30, 40, 50, and 63 μm). (c,d) Same as (a,b), plotted for modification threshold (θ m ) on the y axis. For panels (a-d), the data from Figure 7g (P AMPAR = 600 nm/s) are plotted together for all diameters. (e,f) Same as (a,b), plotted for the AMPAR permeabilities required to achieve a modification threshold of $10 Hz (with the 900-pulses protocol), referred to as threshold P AMPAR , on the y axis. For panels (e,f), the data from Figure 9c are plotted together for all diameters. The insets in panels (e) and (f) depict the inverse of threshold P AMPAR plotted against R in or f 100 , respectively, to illustrate the 1/x relationship between threshold P AMPAR versus R in and threshold P AMPAR versus f 100 . (g,h) Same as (e,f), but plotted threshold P AMPAR was computed to achieve $150% synaptic plasticity with the TBS protocol. For panels (g,h), the data from Figure 10d are plotted together for all diameters.

| DISCUSSION
The principal goal of this study was to assess the mechanistic basis for the expression of plasticity heterogeneities. Plasticity heterogeneity is defined as the variability observed in the amount of plasticity induced by identical activity patterns across cells and synapses of the same subtype. We demonstrate that disparate forms of neural-circuit heterogeneities, spanning intrinsic, synaptic, and structural properties, LTP. Such neuron-to-neuron and animal-to-animal variability in the amount of plasticity induced is typically not analyzed quantitatively, with the data typically represented using summary statistics and interpretations drawn from the average plasticity across different cells from different animals. However, given the role of such differential plasticity across different neurons in resource allocation and in engram formation, it is essential to not just report these heterogeneities but also examine the mechanisms underlying such cell-to-cell differences.
To emphasize the critical roles played by these plasticity heterogeneities across different cells and different synapses, let us consider an extreme scenario where these heterogeneities were absent. This would translate to all synapses across all cells undergoing the same amount of plasticity for any given context, together resulting in the absence of context-dependent recruitment/allocation of synapses or cells that are critical for engram cell formation and decorrelation. From the engram cell formation perspective, there are several lines of evidence to suggest context-dependent plasticity in a subset of cells that are recruited to encode a new context (Josselyn & Frankland, 2018;Josselyn & Tonegawa, 2020;Lau et al., 2020;Lodge & Bischofberger, 2019;Park et al., 2016;Pignatelli et al., 2019;Schmidt-Hieber et al., 2004;Yiu et al., 2014). In addition, afferent connectivity has been demonstrated to be a dominant mediator of neural decorrelation (Mishra & Narayanan, 2019), with strong lines of evidence suggesting that afferent connectivity is actively driven by differences in plasticity profiles across different GCs (Aimone et al., 2006(Aimone et al., , 2009Aimone et al., 2014;Ge et al., 2007;Li et al., 2017;Lodge & Bischofberger, 2019;Luna et al., 2019;Schmidt-Hieber et al., 2004). Thus, in the absence of plasticity heterogeneities, the critical role of differential plasticity in mediating differential connectivity to neurons in the DG during encoding and storage process would be hampered. Our study explores the mechanistic basis for such heterogeneity and traces the potential origins to the pronounced heterogeneities in intrinsic, synaptic, and structural properties of DG GCs. These analyses emphasize the need for studies that assess neural plasticity to quantitatively report plasticity heterogeneities and to trace their origins, under physiological or pathological conditions.  -10). Importantly, this form of degeneracy was demonstrated in a heterogeneous population of neurons that manifested physiologically constrained (Table 2)  paring with previous studies on degeneracy, we note that these studies accounted for degeneracy either in characteristic neuronal intrinsic properties (Mishra & Narayanan, 2019, 2021a, 2021b  profiles is the explosion in the degrees of freedom available for the neurons to achieve these characteristic features, thereby providing multiple routes to achieving functional robustness (Edelman & Gally, 2001;Goaillard & Marder, 2021;Rathour & Narayanan, 2019).
In addition, given the expression of such degeneracy, it is essential that the theoretical and experimental analyses recognize that the mappings between structural components and functional outcomes are many-to-many and avoid reductionist oversimplifications of structure-function relationships (Goaillard & Marder, 2021;Mishra & Narayanan, 2021a, 2021bRathour & Narayanan, 2019   However, it is essential that future studies account for morphological reconstructions of DG GCs with experimentally determined somatodendritic distributions of channels and receptors and assess plasticity profiles for synapses placed at different somato-dendritic locations (Sjostrom & Hausser, 2006). These studies could also specifically employ immature versus mature dendritic morphologies rather than Second, as our focus here was on excitatory synaptic plasticity, we have not incorporated inhibition into our analyses. However, given the DG circuitry that recruits a diverse set of interneurons that impinge along different locations of the somato-dendritic arbor (Amaral et al., 2007;Andersen et al., 2006;Dieni et al., 2013;Elgueta & Bartos, 2019;Freund & Buzsaki, 1996;Houser, 2007), it is essential that the impact of heterogeneities in inhibitory synaptic inputs on plasticity profiles are also assessed in more detail. In this context, there are lines of evidence that the inhibitory neurotransmitter GABA exerts functionally critical excitatory influences on the immature cells, and through the process of maturation shifts to being inhibitory (Chancey et al., 2013;Ge et al., 2006;Heigele et al., 2016).
Thus, future studies that account for inhibition should also assess the impact of this switch in GABAergic impact on immature versus mature GCs and their plasticity profiles.
Third, whereas our cellular-scale analysis has focused on the biophysical and structural heterogeneities as sources of plasticity heterogeneities, there are other potential sources for plasticity heterogeneities. At the molecular scale, it is possible that heterogeneities in the expression of plasticity related molecules (and associated signaling cascades) across synapses and across neurons of the same subtype could mediate plasticity heterogeneities (Josselyn & Frankland, 2018;Park et al., 2016). At the network scale, when multiple neurons are considered, pre-existing afferent and local connectivity onto these neurons could form yet another potential source of plasticity heterogeneities (Josselyn & Frankland, 2018;Josselyn & Tonegawa, 2020).
In addition, there are lines of evidence for a lower overlap in synaptic inputs impinging on immature GCs compared to inputs to mature GCs (Dieni et al., 2016), suggesting a role for afferent heterogeneities in not just regulating output correlations (Dieni et al., 2016;Mishra & Narayanan, 2019, 2021b but also in mediating plasticity heterogeneities. Thus, future studies could expand the analyses of plasticity heterogeneity beyond the cellular scale to encompass network-and molecular-scale components that could drive plasticity heterogeneities. In assessing plasticity heterogeneities at the network scale, it is important that the analyses are built on realistic networks of excitatory and inhibitory neurons receiving physiologically relevant local as well as afferent input activity. As the synapse-localized calcium dynamics are critical mediators of synaptic plasticity, it is important that such analyses are performed on morphologically realistic neuronal models with realistic calcium dynamics and diffusion (Basak & Narayanan, 2018). At the molecular scale, performing realistic simulations would entail precise measurements of the different plasticityrelated signaling molecules in different synapses and assessing intraand inter-neuronal variability in the concentration of these signaling molecules across different synapses. Quantitative signaling cascades could then be built with realistic calcium inputs (Basak & Narayanan, 2018;Bhalla, 2004Bhalla, , 2014Bhalla et al., 2002;Bhalla & Iyengar, 1999) to assess the molecular sources that mediate plasticity heterogeneity across GC synapses.
In extrapolating our conclusions to an in vivo setting involving engram cell formation, it is essential that in vivo activity patterns and other forms of plasticity are considered as well. One route to approach plasticity in a network that incorporate different (intrinsic, synaptic, structural, and afferent) forms of heterogeneities studied here would be to use heterogeneous network models receiving activity patterns from the entorhinal cortices (Mishra & Narayanan, 2019, 2021b. Neuronal models and their connectivity should be constrained by the DG network, with plasticity implemented through the calcium control hypothesis employed here. In a heterogeneous conductance-based setting, calcium through voltage-and ligand-gated calcium channels could contribute to heterogeneous calcium influx across different neurons. The neuronal population could be constructed with mature or immature neurons, with differential connectivity and ionchannel densities to provide insights about plasticity heterogeneities in an in vivo setting. Predictions from such heterogeneous biophysical networks could then be tested in DG networks using in vivo electrophysiology and/or population imaging of calcium activity in awake behaving animals. Furthermore, plasticity in the DG GCs is known to span synaptic and intrinsic properties (Bliss & Lomo, 1973;Lopez-Rojas et al., 2016;Mishra & Narayanan, 2021c. These observations necessitate future studies to assess the impact of neural heterogeneities on conjunctive intrinsic and synaptic plasticity, especially with reference to plasticity heterogeneities, resource allocation, and engram formation (Josselyn & Frankland, 2018;Josselyn & Tonegawa, 2020;Lisman et al., 2018;Mishra & Narayanan, 2021c;Park et al., 2016;Rao-Ruiz et al., 2019;Silva et al., 2009). Analyses of the heterogeneities in such conjunctive plasticity involving multiple components, along with their roles in context-specific resource allocation, could provide crucial insights about how the brain accomplishes stable and continual learning in an ever-changing environment (Mishra & Narayanan, 2021c).
Finally, and importantly, our analyses emphasize the need to systematically characterize the expression of plasticity heterogeneities across different brain regions. Such analyses should span behavioral learning processes and pathological conditions to probe the mechanistic origins of and functional implications for plasticity heterogeneity. For instance, could pathology-induced hyperplasticity that spans several neurological disorders (Bernier et al., 2011;Calabresi et al., 2003;Chattarji et al., 2015;Hulme et al., 2013;Kauer & Malenka, 2007;Markram & Markram, 2010;Rinaldi et al., 2008;Roozendaal et al., 2009;Soda et al., 2019) be a mechanism to reduce plasticity heterogeneity across neurons, thereby hampering context-specific memory formation? Could loss of plasticity heterogeneity in the amygdala be a mechanism behind fear generalization that is observed with certain neurological disorders Markram et al., 2008;Rahman et al., 2017;Suvrathan et al., 2014)?
Could the ability of different neural-circuit components-spanning transmembrane proteins, cytosolic and nuclear signaling elements, synaptic strength, neuronal morphology-to synergistically contribute to similar plasticity profiles (i.e., plasticity degeneracy) provide a therapeutic route for robustness in neural learning hampered by pathological conditions?
These and other associated questions need to be systematically addressed through quantitative characterization of plasticity heterogeneities spanning different brain regions, physiological contexts, and pathological conditions, together assessing the implications for plasticity heterogeneities in context-specific encoding of learned behavior. Mishra, and Rishikesh Narayanan co-wrote the paper.