Disparate forms of heterogeneities and interactions among them drive channel decorrelation in the dentate gyrus: Degeneracy and dominance

Abstract The ability of a neuronal population to effectuate channel decorrelation, which is one form of response decorrelation, has been identified as an essential prelude to efficient neural encoding. To what extent are diverse forms of local and afferent heterogeneities essential in accomplishing channel decorrelation in the dentate gyrus (DG)? Here, we incrementally incorporated four distinct forms of biological heterogeneities into conductance‐based network models of the DG and systematically delineate their relative contributions to channel decorrelation. First, to effectively incorporate intrinsic heterogeneities, we built physiologically validated heterogeneous populations of granule (GC) and basket cells (BC) through independent stochastic search algorithms spanning exhaustive parametric spaces. These stochastic search algorithms, which were independently constrained by experimentally determined ion channels and by neurophysiological signatures, revealed cellular‐scale degeneracy in the DG. Specifically, in GC and BC populations, disparate parametric combinations yielded similar physiological signatures, with underlying parameters exhibiting significant variability and weak pair‐wise correlations. Second, we introduced synaptic heterogeneities through randomization of local synaptic strengths. Third, in including adult neurogenesis, we subjected the valid model populations to randomized structural plasticity and matched neuronal excitability to electrophysiological data. We assessed networks comprising different combinations of these three local heterogeneities with identical or heterogeneous afferent inputs from the entorhinal cortex. We found that the three forms of local heterogeneities were independently and synergistically capable of mediating significant channel decorrelation when the network was driven by identical afferent inputs. However, when we incorporated afferent heterogeneities into the network to account for the divergence in DG afferent connectivity, the impact of all three forms of local heterogeneities was significantly suppressed by the dominant role of afferent heterogeneities in mediating channel decorrelation. Our results unveil a unique convergence of cellular‐ and network‐scale degeneracy in the emergence of channel decorrelation in the DG, whereby disparate forms of local and afferent heterogeneities could synergistically drive input discriminability.

In the DG network, there are at least four distinct forms of heterogeneities that could mediate response decorrelation (the first three are local to the DG network, whereas the fourth is afferent onto the network): (i) heterogeneity in intrinsic ion channel and excitability properties of the neurons; (ii) nonuniformities in the local synaptic connectivity; (iii) structural heterogeneities in neurons introduced by adult neurogenesis; and (iv) input-driven heterogeneity that is reflective of the distinct sets of afferent inputs that impinge on different neurons (as a consequence of the unique divergence in DG connectivity). Which of these distinct forms of heterogeneities play a critical role in mediating channel decorrelation in the DG when they coexpress? What is the impact of cell-to-cell variability in ion channel properties and excitability on channel decorrelation in the DG network receiving different patterns of inputs? Is there a relative dominance among these disparate forms of heterogeneities when they coexpress?
How does the contribution of local network heterogeneities to channel decorrelation change in the presence of unique, sparse, and orthogonal external inputs, an important and unique form of afferent heterogeneity that expresses in the DG network Aimone, Wiles, & Gage, 2006;Aimone, Wiles, & Gage, 2009;Li et al., 2017)?
In this study, we systematically and incrementally incorporate the four different forms of heterogeneities into conductance-based network models of the DG and delineate the impact of each form of heterogeneity on channel decorrelation. Specifically, we used a stochastic search algorithm spanning an exhaustive parametric space (involving experimentally determined ion channel and neurophysiological properties) to reveal cellular-scale degeneracy in the DG, whereby disparate combinations of passive and active properties yielded analogous cellular physiology of excitatory granule (GC) and inhibitory basket cell (BC) populations. Next, we further expanded the parametric search space to encompass biologically observed heterogeneities in local/ afferent network connectivity and in neurogenesis-induced alteration to neuronal structure and excitability. We systematically assessed channel decorrelation in different DG networks, each built with incremental addition of the four distinct forms of heterogeneities. We found that in the absence of afferent heterogeneities, that is, when the DG network was driven by identical afferent inputs, the three forms of local heterogeneities were independently and synergistically capable of mediating significant channel decorrelation. Under these scenarios where the network received identical inputs, we demonstrate a hierarchy of heterogeneities-synaptic, intrinsic, neurogenesisinduced structural, in increasing order of dominance when they coexpress-in effectuating channel decorrelation. Importantly, when we incorporated afferent heterogeneities into the network to account for the unique activity-dependent sparseness and neurogenesis-driven synapse formation in DG afferent connectivity (Aimone et al., 2006;Aimone et al., 2009;Li et al., 2017), we found that the impact of all three forms of local heterogeneities were suppressed by the dominant role played by afferent heterogeneities in mediating the emergence of channel decorrelation. These conclusions point to degeneracy (Edelman & Gally, 2001;, specifically with reference to the emergence of channel decorrelation, with the relative contributions of individual forms of heterogeneities critically regulated by several factors including the degree of divergence of afferent inputs. In elucidating a dominance hierarchy among disparate forms of heterogeneities in terms of their ability to mediate response decorrelation, our results quantitatively demonstrate that the ability of local heterogeneities to decorrelate identical inputs does not necessarily translate to them being effective in decorrelation when different degrees of afferent heterogeneities are present.

| METHODS
The principal goal of this study was to systematically assess the impact of different forms of heterogeneities on response decorrelation in the DG. Our specific focus in this study is on channel decorrelation ( Figure 1a) (one form of response decorrelation that is distinct from pattern decorrelation; Figure 1b), where we assess the correlation between response profiles of individual channels (neurons) to afferent stimuli. Specifically, channel decorrelation decreases the overlap between channel responses, resulting in a code that is efficient because the information conveyed by different channels is largely complementary (Wiechert et al., 2010). In assessing the role of different forms of heterogeneities on channel decorrelation ( Figure 1a), we took advantage of the versatility of conductance-based neuronal network models, and distinguished between four different types of heterogeneities: (i) intrinsic heterogeneity, where the GC and BC model neurons that were used to construct the network had widely variable intrinsic parametric combinations yielding physiological measurements that matched their experimental counterparts.
These heterogeneous model populations were obtained using independent stochastic search procedures for GCs and BCs; (ii) synaptic heterogeneity, where the synaptic strength of the local GC-BC network was variable with excitatory and inhibitory synaptic permeability values picked from uniform random distributions; (iii) neurogenesisinduced heterogeneity in age/structure of the neuron, where the DG network could be made entirely of mature or immature neurons, or be constructed from neurons that represented different randomized neuronal ages; and (iv) input-driven or afferent heterogeneity, where all neurons in the GC and BC populations received either identical inputs (absence of afferent heterogeneity) from the EC, or each GC and BC received unique inputs (presence of afferent heterogeneity) from the EC. The presence of afferent heterogeneity is representative of the sparseness of afferent connections from the EC to the DG, whereby neurons in the DG do not receive the same set of EC inputs during an arena traversal. We present the methodology to account for four different forms of heterogeneities, also providing details on the construction of the network, the measurements, and the analysis techniques used.  & Marder, 2004;Rathour & Narayanan, 2012;Rathour & Narayanan, 2014;Srikanth & Narayanan, 2015), an approach that we refer to as multi-parametric multi-objective stochastic search (MPMOSS), provided us an ideal route to generate a heterogeneous population of GC and BC neuronal models. The choice of this strategy ensured that we have models that are constructed with disparate parameters, but matched with their experimental counterparts in terms of several physiological measurements. In performing MPMOSS on granule cell model parameters, we first tuned a base model that matched with nine different active and passive physiological measurements of granule cells (Figure 2c-g). The passive model parameters of granule cell were as follows: the resting membrane potential (V RMP ), −75 mV; specific membrane resistance, R m = 38 kΩ cm 2 ; and specific membrane capacitance, C m = 1 μF/cm 2 . This allowed us to set the passive charging time constant (R m C m ) to be 38 ms (Schmidt-Hieber, Jonas, & Bischofberger, 2007). Then, to set the passive input resistance (R in ) of the cell to match the experimental value of 309 AE 14 MΩ (Chen, 2004), we set the geometry of the model cell to be a cylinder of 63 μm diameter and 63 μm length (R in = R m /(πdL) = 38 × 10 3 × 10 −2 × 10 −2 /(π × 63 × 10 −6 × 63 × 10 −6 ) = 305 MΩ).

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We introduced nine different active conductances into the GC neuronal model (Santhakumar, Aradi, & Soltesz, 2005): hyperpolarizationactivated cyclic nucleotide gated (HCN or h), A-type potassium (KA), fast sodium (NaF), delayed-rectifier potassium (KDR), small conductance (SK), and big conductance calcium-activated potassium (BK), L-type calcium (CaL), N-type calcium (CaN), and T-type calcium (CaT). FIGURE 1 Two forms of response decorrelation: channel decorrelation and pattern decorrelation. (a) Illustration of channel decorrelation. A trajectory of an animal in Arena 1 results in temporally aligned inputs arriving onto a network of neurons. Individual neurons within the network elicit outputs to these inputs. Channel decorrelation is assessed by computing pair-wise correlations across temporally aligned outputs of individual neurons (channels) within the network, when inputs corresponding to a single pattern (Arena 1) arrive onto the network. Channel decorrelation is computed to determine redundancy in individual neuronal outputs to afferent inputs. (b) Illustration of pattern decorrelation. Two trajectories of an animal in two distinct arenas (Arena 1 and Arena 2) results in distinct sets of inputs arriving onto the same network, at two different time periods T 1 (Arena 1 traversal) and T 2 (Arena 2 traversal). Neurons in the network elicit two sets of outputs (as opposed to the single set of outputs analyzed with reference to channel decorrelation) as the animal traverses Arena 1 or Arena 2. Pattern decorrelation is assessed by computing correlations across these two sets of neuronal outputs when inputs corresponding to two different arenas (patterns) arrive onto the same network. Pattern decorrelation is computed to determine the ability of neuronal outputs to distinguish between the two input patterns (in this case, corresponding to the two arenas). In this study, our focus is on assessing the impact of distinct biological heterogeneities on channel decorrelation [Color figure can be viewed at wileyonlinelibrary.com] The channel kinetics and their voltage-dependent properties were adopted from experimental measurements from the GC (Aradi & Holmes, 1999;Beck, Ficker, & Heinemann, 1992;Ferrante, Migliore, & Ascoli, 2009;Magee, 1998). The reversal potentials for Na, K, and h channels were set as 55 mV, −90 mV, and − 30 mV, respectively. All calcium channels were modeled using the Goldman-Hodgkin-Katz (GHK) formulation (Goldman, 1943;Hodgkin & Katz, 1949), with default values of intracellular and extracellular calcium concentrations set as 50 nM and 2 mM, respectively. The evolution of cytosolic calcium concentration [Ca] c , defining its dependence on calcium current and its decay, was adopted from the formulation (Carnevale & Hines, 2006;Destexhe, Babloyantz, & Sejnowski, 1993;Narayanan & Johnston, 2010;Poirazi, Brannon, & Mel, 2003): where F represented Faraday's constant, τ Ca = 160 ms defined the calcium decay constant in GCs (Eliot & Johnston, 1994), dpt = 0.1 μm was the depth of the shell into which calcium influx occurred, and [Ca] ∞ = 50 nM is the steady-state value of [Ca] c . Conductance-based models of GCs (left) and BCs (right) expressed different sets of ion channels and received external inputs from several MEC and LEC cells. (c-g) The nine physiological measurements used in defining the GC populations: input resistance, R in , measured as the slope of a V-I curve obtained by plotting steady-state voltage responses to current pulses of amplitude −50 to 50 pA, in steps of 10 pA, for 500 ms (c); sag ratio, measured as the ratio between the steady-state voltage response and the peak voltage response to a −50 pA current pulse for 1 s (d); firing rate in response to 50 pA, f 50 (c) and 150 pA current injection, f 150 (e); spike frequency adaptation (SFA) computed as the ratio between the first (ISI first ) and the last (ISI last ) interspike intervals in spiking response to a 150 pA current injection (e); action potential half-width, T APHW (f ); action potential threshold, computed as the voltage at the time point where dV m /dt crosses 20 V/s (f ); action potential amplitude, V AP (g) and the fast after hyperpolarization potential (V AHP In performing MPMOSS on this GC base model, we used a search space spanning 38 active parameters associated with the nine active conductances and two parameters that defined the passive properties of the model (leak conductance, g L = 1/R m and C m ). We generated 20,000 unique models by randomly picking the values of 40 parameters from independent uniform distributions that spanned the range for that specific parameter (Table 1). The multiple objectives of this MPMOSS strategy was with reference to bounds on nine different measurements computed for each of these 20,000 models, and the goal was to find models that had all nine measurements fall within their experimentally set bounds (Table 2). We found 126 models (~0.63% of the total population) to be valid in terms of achieving these multiple objectives, which were used as the heterogeneous GC population.
A similar MPMOSS strategy was used to generate a heterogeneous population of basket cells, whose geometry was set as a cylinder with 66 μm diameter and 66 μm length. The passive parameters of the BC base model were as follows: V RMP = −65 mV, R m = 7.1 kΩ cm 2 , C m = 1 μF/cm 2 . Four different voltage-gated ion channels (HCN, KA, NaF, and KDR) were introduced into the model, with the parameters set to match experimental measurements (Magee, 1998;Santhakumar et al., 2005). With these passive and active parametric values (Table 3), the R in of the base BC model was 57 MΩ.
The stochastic search for BCs involved 16 parameters associated with the four voltage-gated ion channels and two parameters defining passive membrane properties. Together, we picked 18 passive and active parametric values from independent uniform distributions (bounds are shown in Table 3), and generated 8,000 unique BC models. The physiological measurements that constituted the multiple objectives in defining the validity of BC models were the same as those for GCs, but with different experimentally derived ranges for each measurement (Table 4). This procedure yielded 54 valid BC models (~0.675% of the total population) with significant heterogeneity in each of the 18 intrinsic parameters that constructed them, and were used as the heterogeneous BC population. The experimental bounds on measurements for granule (Table 2) and basket (Table 4) cells were obtained from (Aradi & Holmes, 1999;Krueppel, Remy, & Beck, 2011;Lubke, Frotscher, & Spruston, 1998;Mott, Turner, Okazaki, & Lewis, 1997;Santhakumar et al., 2005).    (Boss, Peterson, & Cowan, 1985). Although the default net- connectivity was set such that the probability of a BC to GC connection was 0.1, and that of a GC to BC connection was set as 0.05 (Aimone et al., 2009).
The GC ! BC and BC ! GC connections were modeled as synapses containing AMPA and GABA A receptors, respectively. The GC ! BC AMPA receptor current as a function of voltage (v) and time (t) was modeled, following the GHK convention (Goldman, 1943;Hodgkin & Katz, 1949;Narayanan & Johnston, 2010): where, where F is the Faraday's constant, R is the gas constant, T is the temperature and P AMPAR is the maximum permeability of AMPAR. s(t) governed the AMPAR kinetics and was set as follows: where a normalized s(t) such that 0 ≤ s(t) ≤ 1, τ d (= 10 ms) represented the decay time constant, τ r (= 2 ms) depicted the rise time, where P GABAAR was the maximum permeability of GABA A receptor. s Simulations were performed for various combinations of synaptic permeability parameters P AMPAR and P GABAAR . These parameters were maintained at a regime where the peak-firing rate of GCs and BCs stayed within their experimental ranges of 4-10 Hz and 30-50 Hz, respectively (Leutgeb et al., 2007). We ensured that extreme parametric combinations where the cell ceased firing (because of depolarization-induced block at one extreme or high inhibition at the other) were avoided. When homogeneous synaptic connectivity was used, all P AMPAR and P GABAAR were set to identical values across the network, with different sets of network simulations performed with different P AMPAR -P GABAAR combinations ( Figure 7b). In introducing local synaptic heterogeneity, we picked ranges for P AMPAR and P GABAAR that satisfied the firing rate requirements above and picked values for P AMPAR and P GABAAR (for all synapses in the network) from independent uniform distributions spanning this range ( Figure 7c).

| Input-driven afferent heterogeneities: External inputs from the entorhinal cortex
where (x, y) represented the position of the virtual animal in the arena, and g 1 , g 2 , and g 3 were defined as FIGURE 7 Heterogeneities in the strength of local network connections modulate channel decorrelation, with increase in inhibitory synaptic strength enhancing network decorrelation. (a) Lower triangular part of correlation matrix representing pair-wise Pearson's correlation coefficient computed for firing rates of 500 GCs. Note that there was no heterogeneity in the synaptic strengths of local connections, with AMPAR and GABAAR permeability across local network synapses set at fixed values. Shown are four different correlation matrices, with P AMPAR (1 or 5 nm/s) and P GABAAR (10 or 50 nm/s) fixed at one of the two values. (b) Left, cumulative distribution of correlation coefficients for firing rates of 500 GCs, computed when the simulations were performed with different sets of fixed values of P AMPAR (spanning 1-5 nm/s) and P GABAAR (spanning 10-50 nm/s). The gray-shaded plots on the extremes were computed from corresponding matrices shown in (a). Right, cumulative distributions of correlation coefficients corresponding to the gray-shaded plots on the left, to emphasize the impact of synaptic heterogeneity on decorrelation. (c) Distribution of P AMPAR and P GABAAR in a network of heterogeneous GC and BC populations, constructed with heterogeneity in local synaptic strengths as well. Each AMPA and GABA A receptor permeability was picked from a uniform distribution that spanned the respective ranges. The color codes of arrows and plots correspond to cases plotted in (d,e). (d) Lower triangular part of correlation matrices representing pair-wise Pearson's correlation coefficient computed for firing rates of 500 GCs. For the right and left matrices, which are the same plots as in Figure 6c,e, respectively, there was no synaptic heterogeneity, with P AMPAR and P GABAAR set at specified fixed values for all excitatory and inhibitory synapses. The matrix represented in the center was computed from a network endowed with intrinsic and synaptic heterogeneity (shown in c). (e) Cumulative distribution of correlation coefficients represented in matrices in (d). Plotted are distributions from five different trials of each configuration. Note that except for the homogenous population, all three configurations were endowed with intrinsic heterogeneity. The configurations "intrinsic + synaptic heterogeneity" and "homogeneous + synaptic heterogeneity" had randomized synaptic permeabilities; for the other two configurations, the synaptic strengths were fixed at specific values: high P, P AMPAR = 5 nm/s, and P GABAAR = 40 nm/s; low P, P AMPAR = 1 nm/s, and P GABAAR = 20 nm/s [Color figure can be viewed at wileyonlinelibrary.com] where λ represents the grid frequency, θ represents the grid orientation, and x 0 , y 0 were offsets in x, y, respectively. This hexagonal grid function was scaled to obtain the input from a single MEC cell  In the default network (500 GC and 75 BC cells), correlation matrices for the GCs (500 × 500) were constructed by computing Pearson's correlation coefficient of respective instantaneous firing rate arrays (each spanning 1,000 s). Specifically, the (i, j)th element of these matrices was assigned the Pearson's correlation coefficient computed between the instantaneous firing rate arrays of neuron i and neuron j in the network (to assess channel decorrelation; Figure 1a). As these correlation matrices are symmetric with all diagonal elements set to unity, we used only the lower triangular part of these matrices for analysis and representation. In assessing channel decorrelation, irrespective of the specific set of heterogeneities incorporated into the network, we first plotted the distribution of these correlation coefficients. In addition, we represented correlation coefficients from individual distributions as mean AE SEM, and used the Kolmogorov Smirnov test to assess significance of differences between distributions.
In assessing channel decorrelation as a function of input correlation, we first computed the total afferent current impinging on each neuron. As the total current was the same for scenarios where identi-

| Computational details
All simulations were performed using the NEURON simulation environment (Carnevale & Hines, 2006), at 34 C with an integration time step of 25 μs. Analysis was performed using custom-built software written in Igor Pro programming environment (Wavemetrics). Statistical tests were performed in statistical computing language R (www.Rproject.org).

| RESULTS
In systematically delineating the impact of distinct forms of heterogeneities on channel decorrelation (Figure 1a), we constructed networks

| Degeneracy in single neuron physiology of granule and basket cell model populations
We used a well-established stochastic search strategy (Foster et al., 1993;Goldman et al., 2001;Prinz et al., 2004;Rathour & Narayanan, 2014) to arrive at populations of conductance-based models for GCs and BCs. This exhaustive and unbiased parametric search procedure was performed on 40 parameters for GCs (Table 1), and 18 parameters for BCs (Table 3), involving ion channel properties derived from respective neuronal subtypes. Nine different measurements, defining excitability and action potential firing patterns ( Figure 2 and Table 2), were obtained from each of the 20,000 stochastically generated unique GC models, and were matched with corresponding electrophysiological GC measurements. We found 126 of the 20,000 models (~0.63%) where all nine measurements were within these Note that all three configurations were endowed with intrinsic heterogeneities (heterogeneous GC and BC populations), and all cells in the network received identical external inputs. The "intrinsic + synaptic heterogeneity" configuration had randomized synaptic permeabilities; for the other two configurations, the synaptic strengths were fixed at specific values: high P, P AMPAR = 700 nm/s, and P GABAAR = 70 nm/s; low P, P AMPAR = 7 nm/s, and P GABAAR = 9 nm/s. (c) Firing rates, represented as quartiles, of all the GCs plotted for the different networks (heterogeneous vs identical input case) they resided in. (d) Cumulative distribution of correlation coefficients of firing rates computed from granule cell firing rates in networks constructed with different forms of age-related heterogeneities (fully immature, fully mature and variable age). Panels on the top and bottom respectively correspond to networks receiving identical and heterogeneous external inputs from the EC. All three populations were endowed with intrinsic and synaptic heterogeneities.  (Table 2), and thus were declared as valid GC models. A similar procedure was used for BC cells, where 9 different measurements from 8,000 unique models were compared with corresponding electrophysiological BC measurements. Here, we found 54 of the 8,000 models (~0.675%) where all nine measurements were within electrophysiological bounds (Table 4), and declared them as valid BC models. The experimental bounds on physiological measurements for granule (Table 2) and basket (Table 4) cells were obtained from references (Aradi & Holmes, 1999;Krueppel et al., 2011;Lubke et al., 1998;Mott et al., 1997;Santhakumar et al., 2005). How did these neuronal populations achieve degeneracy? Did they achieve this by pair-wise compensation across parameters, or was change in one parameter compensated by changes in several other parameters to achieve robust physiological equivalence? In answering this, we plotted pair-wise scatter plots, independently on neurons with lower excitability (e.g., Cell #2 in Figure 6a), but would be suprathreshold for neurons with relatively higher excitability (e.g., Cell #5 in Figure 6a), thereby manifesting as changes in firing rate or in the emergence of place fields at specific locations (Lee, Lin, & Lee, 2012). These observations suggest that DG neurons could undergo rate remapping (Leutgeb et al., 2007;Renno-Costa et al., 2010) merely as a consequence of plasticity in intrinsic excitability. To introduce neurogenesis-induced heterogeneity into our network, we noted that the excitability of new born neurons in the DG, which could mature to either GCs or BCs, is higher as a consequence of lower surface area reflective of the diminished arborization of immature neurons (Aimone et al., 2014;Liu et al., 2003;Schmidt-Hieber et al., 2004;Wang et al., 2000). To quantitatively match the excitability properties of these neurons, we introduced structural plasticity by reducing the surface area of the valid GC and BC models ( Figure 8) through reduction of their diameter. This reduction in surface area expresses as an increase in input resistance (Esposito et al., 2005;Rall, 1977;Schmidt-Hieber et al., 2004) in each of these neurons (Figure 8a), which in turn translates to increase in firing rate ( Figure 8b).
With the ability to introduce intrinsic, synaptic, and neurogenesisinduced forms of heterogeneity into our network, we analyzed three distinct networks (fully mature, fully immature, and variable age) to specifically understand the role of neurogenesis-induced heterogeneity on channel decorrelation to identical inputs. All three networks were endowed with intrinsic and synaptic heterogeneities receiving afferent inputs from the same arena (Figures 6 and 7), and the distinction between the three cases was only with reference to neuronal age ( Figure 8d). In comparing the firing rates of the GCs for different network configurations, we found that the firing rates of all GCs were comparable for all cases where neurogenesis-induced heterogeneities were absent. However, with the introduction of neurogenesis, especially in the scenario where all cells were immature, the firing rates increased and spanned a larger range. In the more physiologically relevant scenario of heterogeneous cellular age, although the firing rates spanned a larger range, a significant proportion of them were in the low firing regime characteristic of GCs (Figure 8e).
We found that the level of channel decorrelation in the fully immature network was significantly (KS test; p < .001) higher than that achieved in the fully mature network (Figure 8f ). This is to be expected because the structural heterogeneity (effectuated by changes in diameter) would amplify the inherent intrinsic heterogeneity of neurons in the network, thereby enhancing the beneficiary effects of intrinsic heterogeneity that we had observed earlier ( Figure 6). Importantly, reminiscent of our results with the introduction of synaptic heterogeneity (Figure 7), in the network that was endowed with variability in neuronal age, the level of decorrelation was intermediate between that obtained with the fully mature and the fully immature networks (Figure 8f ). Together, these results demonstrate that neurogenesis-induced variability in neuronal response properties adds an additional layer of structural heterogeneity in the DG network, and enhances channel decorrelation to identical external inputs. These results also demonstrate that among the three local heterogeneities assessed thus far, neurogenesis-induced structural heterogeneities form the dominant heterogeneity, capable of inducing a much larger response decorrelation (compared to the input correlation set at 1, consequent to the identical nature of afferent inputs) compared to either synaptic or baseline intrinsic heterogeneities.
Together, our experimental design involving systematic incorporation of biophysical, synaptic, and structural heterogeneities allowed us to specifically demonstrate a hierarchy of heterogeneities-synaptic, intrinsic, and neurogenesis-induced structural, in increasing order of dominance when they coexpress-in effectuating channel decorrelation.
3.5 | Input-driven heterogeneity mediated by sparseness of afferent connectivity is a dominant regulator of channel decorrelation An important defining characteristic of the rodent DG network is the sparseness of the afferent connectivity matrix that is reflective of massive convergence and divergence reflecting the small (~94,000) number of layer II EC cells (Gatome, Slomianka, Lipp, & Amrein, 2010;Mulders, West, & Slomianka, 1997) that project to a large (~1.2 million) number of DG cells (Rapp & Gallagher, 1996;West, Slomianka, & Gundersen, 1991), resulting in unique, sparse, and orthogonal set of afferent external inputs impinging on each GC Anderson et al., 2007;Li et al., 2017). Thus far in our analysis, in an effort to delineate the impact of three distinct forms of heterogeneity, we used an artificial construct where all neurons in the network received identical inputs. To assess the impact of this fourth form of afferent input-driven heterogeneity, we introduced divergence in the set of EC neurons that project onto each GC and BC. This implied that each GC and BC now received distinct sets of EC inputs.
As a consequence of distinct set of inputs impinging on each GC, the firing fields were distinct across different GCs (Figure 9a), unlike the near-identical firing fields (except for differences in firing frequency or threshold for firing) in the case where neurons received identical inputs (Figure 6a). Importantly, when we analyzed pair-wise correlation of firing rates across different neurons, we found that the correlation coefficients were lower irrespective of the presence or absence of different forms of heterogeneity (Figures 9b and 10b). The overall firing rate distributions obtained with either identical When we plotted the cumulative distributions of correlation coefficients obtained with the introduction of distinct forms of local network heterogeneities, we found them to significantly overlap with each other (Figure 9d). This is in stark contrast to the network receiving identical external inputs (Figures 7e and 8f ), where introduction of each of intrinsic, synaptic, and neurogenesis-induced heterogeneities enhanced or altered the level of response decorrelation, with a welldefined hierarchy among these heterogeneities (Figures 6-8). The where there were more afferent inputs arriving into the network. As a next step in our sensitivity analyses, we asked if our conclusions on the role of different forms of heterogeneities were scalable and were invariant to network size? To test this, we repeated our analyses in There is a growing body of evidence that suggests that the high excitability of immature GCs in the DG is counterbalanced by lower synaptic drive (Dieni et al., 2013;Dieni et al., 2016;Li et al., 2017;Mongiat et al., 2009). To accommodate this into our model and test the impact of such counterbalance on our conclusions, we rescaled the synaptic drive to immature neurons in an excitability-dependent manner such that the variability in firing rate was reduced (Figure 13b; cf. Figures 10a and 12c). In addition, as lines of evidence for adult neurogenesis in BCs are not as broad as that for GCs, we asked if our conclusions on the dominant role of afferent heterogeneities would hold if adult neurogenesis were restricted only to GCs with the BC population remaining mature. When we repeated our analyses with several distinct configurations involving rescaled inputs and with structural heterogeneities associated with adult neurogenesis confined only to GCs, we found the dominance of afferent heterogeneities and the relative dominance among local heterogeneities when presented with identical afferent stimuli to hold even under these conditions ( Figure 13). In addition, we also found that the higher levels of input-output decorrelation that was observed in the purely immature populations receiving heterogeneous inputs (Figures 10c,d, 11d, and 12f ) was not observed when this population received rescaled synaptic drive (Figure 13d). This suggested that the apparent increase in the specific levels of decorrelation that was observed earlier was a mere reflection of the huge variability in the firing rates. When this fir- is that if local network heterogeneities can elicit significant response decorrelation even for identical inputs, the contribution of these local network heterogeneities would be sustained or amplified when the inputs become heterogeneous. In other words, the implicit assumption is that there would be significant contributions from local network heterogeneities even under more realistic conditions where the inputs are unique, sparse, and orthogonal (Li et al., 2017). The conclusions of this study instead demonstrate that the contributions of local network heterogeneities are significantly suppressed (not enhanced or sustained) when an additional and physiologically critical (Li et al., 2017) layer of afferent heterogeneities coexpresses in the network.

| DISCUSSION
Adult neurogenesis in the DG has been shown to drive unique, sparse, and orthogonal afferent inputs onto DG neurons (Li et al., 2017), which are postulated to subserve efficacious information transfer by reducing neuronal response correlations (Chow et al., 2012;Padmanabhan & Urban, 2010;Pitkow & Meister, 2012;Tetzlaff et al., 2012;Wiechert et al., 2010).  (Hanus & Schuman, 2013;Mittal & Narayanan, 2018;Rathour & Narayanan, 2012;Rathour & Narayanan, 2014). Second, with reference to networks of neurons that received identical inputs, our analyses showed that heterogeneities in intrinsic neuronal properties and local synaptic heterogeneities (including local synaptic inhibition) could drive response decorrelation across neurons, either individually or synergistically when they are expressed together. These analyses also presented a hierarchy of local heterogeneities in mediating response decorrelation, whereby intrinsic heterogeneities were the dominant form between intrinsic and synaptic heterogeneities. In addition, we demonstrated that neurogenesis-induced structural heterogeneities further enhance the ability of the network to perform input discriminability ( Figures 6-8).
Third, these results also emphasize a potential role for changes in intrinsic neuronal properties as a putative mechanism for rate remapping in granule cells, whereby the rate of firing at a given place field location could be significantly modulated by changes in the intrinsic excitability of the cell, even when the afferent inputs remained the same (Figure 6a). Fourth, incorporating afferent heterogeneities to reflect the specific connectivity pattern and the active recruitment of heterogeneous afferents by the DG network (Andersen, Morris, Amaral, Bliss, & O'Keefe, 2006;Li et al., 2017), we found that the quantitative contributions of local heterogeneities to the emergence of channel decorrelation significantly diminished in the presence of afferent heterogeneities (Figures 10-12). These results imply that with reference to the dentate gyrus endowed with the expression of afferent heterogeneities and sparse connectivity, analyses on response decorrelation should not merely rely on extrapolations from conclusions derived from scenarios with identical inputs.   (Aimone et al., 2006;Aimone et al., 2009;Aimone et al., 2014;Li et al., 2017). In such a scenario, the afferent heterogeneities would be driven by active assignment of spatial connectivity from the EC to individual DG neurons, whereby the novel contexts encountered by the animal are encoded by the temporal onset of neurons. Such active assignments could be driven by activity-dependent connectivity aided by the hyperplastic, hyperexci-  (Aimone et al., 2006;Aimone et al., 2009;Aimone et al., 2014;Alvarez et al., 2016;Dupret et al., 2007;Kropff et al., 2015;Li et al., 2017;Marin-Burgin et al., 2012;Schmidt-Hieber et al., 2004;Tashiro et al., 2006). In addition to this, our results where distinct structural components could combine to elicit analogous function. Given the several possible routes through which similar function can be achieved, it is possible for biological systems to invoke disparate mechanisms to achieve the same function through very different parametric combinations (Edelman & Gally, 2001;Foster et al., 1993;Goldman et al., 2001;Prinz et al., 2004;Rathour & Narayanan, 2014;. In systems that are responsible for encoding of novel information, robust homeostasis of output constitutes only one side of the overall physiological goals   Finally, this computational study further strengthens the need for engaging (and explicitly modeling) different components of a physiological system, involving emergent properties and degeneracy at each scale, to effectively address questions that require synergistic interactions between components at multiple scales. This study also emphasizes the need to individually account for the disparate biological heterogeneities (and nontrivial interactions among them), that are ubiquitously prevalent in neuronal systems, in assessing physiological processes (Anirudhan & Narayanan, 2015;Das, Rathour, & Narayanan, 2017;Goldman et al., 2001;Marder, 2011;Marder & Goaillard, 2006;Marder & Taylor, 2011;Mittal & Narayanan, 2018;Mukunda & Narayanan, 2017;Prinz et al., 2004;Rathour & Narayanan, 2012;Rathour & Narayanan, 2014;Srikanth & Narayanan, 2015).

| Limitations of the analyses and future directions
Although we had systematically incorporated and assessed the role of several forms of heterogeneities into our conductance-based network models, the analyses presented here have limitations, several of which have their origins traceable to the computational complexity associated with putting together a heterogeneous conductance-based network where neurons were endowed with several ion channels. From the perspective of sparse active connectivity that is observed in decorrelating circuits, we had used lesser number of synaptic inputs.
Although this limitation was partly rectified by our simulations with more number of active inputs (Figure 11), future studies could theoretically and experimentally assess the impact of sparseness and heterogeneities in number of synapses towards achieving input discriminability (Cayco-Gajic et al., 2017;Li et al., 2012). In addition, although we did specific analyses addressing the question on scalability (with reference to network size) of our conclusions (Figure 12), it was on a network that was smaller in size. Future studies could extend our analyses by systematically incorporating the several heterogeneities used here into larger networks (Dyhrfjeld-Johnsen et al., 2007;Schneider, Bezaire, & Soltesz, 2012) and assessing if the conclusions are scalable.
We had not modeled or incorporated other cell types within the dentate gyrus into our network model. These other cells include the mossy cells with their unique ability to mediate feedback projections from the CA3, the molecular layer perforant path-associated (MOPP) cells and other interneurons that are prevalent within the DG (Amaral, Scharfman, & Lavenex, 2007;Li et al., 2012;Scharfman & Myers, 2012). It would be of interest for future studies to ask if the afferent heterogeneities are still dominant even if the other cells and associated local heterogeneities express within the dentate circuit. We believe that our conclusions on the dominance of afferent heterogeneities would still hold because of the several lines of sensitivity analyses presented here, and because the incorporation of afferent heterogeneities was into GC cells, the primary cell type of the dentate gyrus, and is based on strong experimental and theoretical lines of evidence (Aimone et al., 2006;Aimone et al., 2009;Aimone et al., 2014;Li et al., 2017). Furthermore, in this study, we generated immature model cells by altering only the structural parameters with constraints on input resistance as a physiological measurement. Future computational studies could employ stochastic search strategies specific to precise morphological reconstructions of immature neurons (Beining et al., 2017a;Beining, Mongiat, Schwarzacher, Cuntz, & Jedlicka, 2017b), coupled with rigorous electrophysiological characterization of their channels, to incorporate age heterogeneity into model populations.
Although our focus in this study was on channel decorrelation ( Figure 1a) in the dentate gyrus, future studies could assess the impact of the disparate set of heterogeneities analyzed here on pattern decorrelation ( Figure 1b). Specifically, whereas channel decorrelation deals with reducing redundancy across output profiles of individual channels (neurons), pattern decorrelation enables neuronal representations of input patterns to be more distinct (Figure 1b), thereby allowing efficient classification of input patterns (Wiechert et al., 2010). These studies could involve distinct arenas where the animal traverses, and assess the impact of morphed arenas presented to neuronal structures in the model (Leutgeb et al., 2007;Renno-Costa et al., 2010). Additionally, conductance-based models with realistic biophysical models of ion channels provide the ability to assess the impact of distinct ion channels on pattern and channel decorrelations. Future computational studies could focus on the specific contribution of different channels to pattern and channel decorrelations within the framework of degeneracy presented here employing the virtual knockout framework (Anirudhan & Narayanan, 2015;Mittal & Narayanan, 2018;Mukunda & Narayanan, 2017;Rathour & Narayanan, 2012;Rathour & Narayanan, 2014), and test predictions from these simulations using pharmacological agents in electrophysiological and behavioral experiments.
In this study, simplified single compartmental models for both GCs and BCs are used to avoid computational complexities associated with networks of morphologically realistic models. However, given the critical role of DG dendritic structures in input integration and discriminability (Chavlis, Petrantonakis, & Poirazi, 2017;Krueppel et al., 2011;Schmidt-Hieber et al., 2007), it is essential to expand our analyses to morphologically realistic conductance-based DG model with differential spatial distributions of MEC and LEC inputs. Such models would also provide an additional layer of morphological heterogeneity (even among mature GC neurons) in dendritic branching patterns. The interactions of the four forms of heterogeneities with the morphological heterogeneity could then be assessed with reference to different forms of response decorrelation in the DG. These analyses, including the role of heterogeneities in dendritic processing in DG neurons in effectuating channel decorrelation or pattern separation (Chavlis et al., 2017) could be assessed using these multi-compartmental single neuron models that are built with biological dendritic heterogeneities incorporated into them (Rathour & Narayanan, 2014). In this context, a recent study presents an updated model of GC also introducing a toolbox named T2N that is an interface between NEURON, MATLAB and TREES (Beining et al., 2017b). This toolbox-in conjunction with the MPMOSS approach, spanning morphology, and channel distribution of immature and mature neurons-forms an ideal substrate to address these questions in large-scale network models endowed with morphological heterogeneity as well (Beining et al., 2017b). Finally, our analyses also predict that rate remapping in DG neurons could also be achieved through plasticity of intrinsic excitability (Figure 6a).
This could be tested directly by asking questions about whether intrinsic plasticity in the DG could mediate rate remapping, and assessing differences in the expression of intrinsic plasticity in mature versus immature neurons, especially given the well-established differences in synaptic plasticity profiles between mature and immature neurons.