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.

Keywords: adult neurogenesis; computational model; degeneracy; hippocampus; parametric variability; sparse connectivity.

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]

Figure 2

Model components and measurements. (a) Schematic representation of the cylindrical neuropil of 156 μm diameter and 40 μm height (left) with the top view (right) showing the distribution of 500 GCs (black) and 75 BCs (red). (b) 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). (h) Inputs from MEC (top) were modeled as grid structures with randomized scale and orientation, whereas inputs from LEC (bottom), carrying contextual information, were represented as smoothed and randomized matrices comprised of active and inactive boxes. Schematic color‐coded representations of individual inputs (5 MEC and 5 LEC cells) and their summations (separate for MEC and LEC inputs) are superimposed on the virtual animal trajectory in an arena of size 1 m × 1 m. (i) Sample GC voltage trace in response to total MEC (top) and LEC (bottom) current inputs. (j) Color‐coded rate map obtained by superimposing firing rate output from an isolated GC in response to both MEC and LEC inputs, as the virtual animal traverses the arena [Color figure can be viewed at wileyonlinelibrary.com]

Figure 3

Illustration of cellular‐scale degeneracy in granule cell physiology with six randomly chosen valid models, where analogous functional characteristics were achieved through disparate parametric combinations. (a) Firing pattern of six randomly chosen valid GC models in response to 150 pA current injection with corresponding measurement values for action potential amplitude (V _AP), action potential half‐width (T _APHW), action potential threshold (V _th), fast after hyperpolarization (V _fAHP), and spike frequency adaptation (SFA). (b) Voltage traces of six valid GC models in response to −50 and 50 pA current injection, with associated measurement values for input resistance (R _in) and sag ratio. Note that firing rate at 150 pA, f ₅₀, was zero for all models. (c) Firing frequency plots for six valid GC models in response to 0–400 pA current injections, indicating values of firing rate at 150 pA for each valid model. Note that all the 9 different measurements are very similar across these six models. (d) Distribution of the 40 underlying parameters in the six valid models, shown with reference to their respective min–max ranges. The color code of the dots is matched with the plots and traces for the corresponding valid models in a–c [Color figure can be viewed at wileyonlinelibrary.com]

Figure 4

Illustration of cellular‐scale degeneracy in basket cell physiology with six randomly chosen valid models, where analogous functional characteristics were derived from disparate parametric combinations. (a) Firing pattern of six randomly chosen valid BC models in response to 150 pA current injection with corresponding measurement values for action potential amplitude (V _AP), action potential half‐width (T _APHW), action potential threshold (V _th), fast after hyperpolarization (V _fAHP), and spike frequency adaptation (SFA). (b) Voltage traces of six valid BC models in response to −50 and 50 pA current injection, with associated measurement values for input resistance (R _in) and sag ratio. (c) Firing frequency plots for six valid BC models in response to 0–800 pA current injections, indicating values of firing rate at 150 pA for each valid model. (d) Distribution of underlying 18 parameters in the six valid BC models, shown with reference to their respective min–max ranges. The color code of the dot is matched with the plots and traces for the corresponding valid model in a–c [Color figure can be viewed at wileyonlinelibrary.com]

Figure 5

Independently heterogeneous populations of granule and basket cells exhibited cellular‐scale degeneracy with weak pair‐wise correlations of underlying parameters. (a) Left, lower triangular part of a matrix comprising pair‐wise scatter plots between 40 parameters underlying all valid GC models (n = 126). The bottom‐most row represents the histograms for corresponding parameters in the valid model population, showing all parameters spanning their respective min–max ranges. Right, upper triangular part of a matrix comprising pair‐wise scatter plots between 18 parameters underlying all valid BC models (n = 54). The topmost row represents the histograms for corresponding parameters in the valid model population, showing all parameters spanning their respective min–max ranges. The red scatter plots indicate that the value of correlation coefficient for the pair was >0.5, whereas the blue scatter plots denote pairs where the correlation coefficient value was <−0.5. (b) Top, heat map of correlation coefficient values for GC cells, corresponding to each scatter plot box depicted in a. Bottom, distribution of correlation coefficient values for the 780 unique pairs, of the 40 parameters, corresponding to scatter plots for GC parameters shown in a. (c) Same as (b) but for BC cells with 153 unique pairs of correlation coefficients (a) [Color figure can be viewed at wileyonlinelibrary.com]

Figure 6

Heterogeneity in intrinsic neuronal excitability is a robust mechanism for achieving channel decorrelation through rate remapping of cellular responses. (a) Voltage traces (left), instantaneous firing rate (middle), and color‐coded rate maps (right; superimposed on the arena) for five different GCs in a network made of a heterogeneous GC and BC populations. (b) Lower triangular part of correlation matrix representing pair‐wise Pearson's correlation coefficient computed for firing rates of 500 GCs spanning the entire 1,000 s simulation period. Inset represents the histogram of these correlation coefficients. Note that there was no heterogeneity in the synaptic strengths of local connections, with P _AMPAR = 5 nm/s and P _GABAAR = 40 nm/s for all excitatory and inhibitory synapses, respectively. (c) Cumulative distribution of correlation coefficients represented in matrix in b. Plotted are distributions from five different trials of the simulation, with each trial different in terms of the cells picked to construct the network. (d,e) Same as (b,c), but with the synaptic strengths of local connections fixed at lower permeability values: P _AMPAR = 1 nm/s and P _GABAAR = 20 nm/s [Color figure can be viewed at wileyonlinelibrary.com]

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]

Figure 8

Incorporation of neurogenesis‐induced structural heterogeneity in neuronal age enhances channel decorrelation in a network of neurons receiving identical inputs. (a) Input resistance of the 126 GCs (left) and 54 BCs (right) plotted as a function of diameter of cell. Dotted lines represent the range for immature cell diameters (2–9 μm for GC and 1–3 μm for BC), obtained from ranges of experimentally obtained input resistance values in immature cells. (b) Firing frequency plotted as a function of diameter in response to 10 pA (closed triangles) and 100 pA (open circles) current injections into the 126 GCs (left) and 54 BCs (right). (c) Distribution of GC (top) and BC (bottom) diameters in a network of heterogeneous GC and BC populations, constructed with heterogeneity in local synaptic strengths and in the age of the neurons. The diameter of each GC and BC in the network was picked from a uniform distribution that spanned respective ranges. The color codes of arrows and plots correspond to fully mature network (green; large diameters), fully immature network (orange; small diameters), and mixed network (purple; variable diameters) cases plotted in (d–f). (d) Lower triangular part of correlation matrices representing pair‐wise Pearson's correlation coefficient computed for firing rates of all GCs. The matrix corresponding to the fully mature population is the same as that in Figure 7d, with the same color code. Note that all three networks were endowed with intrinsic and synaptic heterogeneity, with changes only in the neuronal age. (e) Firing rates, represented as quartiles, of all GCs plotted for the different networks they resided in. (f) Cumulative distribution of correlation coefficients represented in matrices in (d). Plotted are distributions from five different trials of each configuration [Color figure can be viewed at wileyonlinelibrary.com]

Figure 9

Heterogeneous external connectivity is the dominant form of variability that drives channel decorrelation in a network endowed with intrinsic, synaptic, and age heterogeneities. (a) Instantaneous firing rate (left) and color‐coded rate maps (right; superimposed on the arena) for 10 different GCs in a network endowed with intrinsic, synaptic, age, and input‐driven forms of heterogeneities. (b) Lower triangular part of correlation matrices representing pair‐wise Pearson's correlation coefficient computed for firing rates of all GCs. The four different matrices correspond to networks endowed with different sets of heterogeneities. (c) Firing rates, represented as quartiles, of all the GCs plotted for the different networks they resided in. Color codes for the specific set of heterogeneities included into the network are the same as those in Panel b above. (d) Cumulative distribution of correlation coefficients represented in matrices in (b) [Color figure can be viewed at wileyonlinelibrary.com]

Figure 10

Afferent heterogeneities dominate channel decorrelation when they are coexpressed with other local‐network heterogeneities. (a) Firing rates, represented as quartiles, of all the GCs plotted for the different networks (heterogeneous vs identical input) they resided in. Color codes for the specific set of heterogeneities incorporated into the network are the same as those in Figure 9b. (b) Statistical (mean ± SEM) comparison of correlation coefficients obtained with networks, endowed with distinct forms of heterogeneities, receiving identical (solid boxes; derived from Figure 8f) versus variable (open boxes; derived from Figure 9d) external inputs. (c) Response (output) correlation plotted as a function of input correlation. Output correlations are the same as those plotted in Figure 8f (identical inputs) and Figure 9d (heterogeneous inputs). The corresponding input correlations represented Pearson's correlation coefficients computed for afferent current inputs onto individual neurons as the virtual animal traversed the arena. Note that the input correlation for identical input case is 1 with mean output correlation plotted correspondingly for identical case. (d) The difference between input correlation and respective output correlation (for individual pairs of neurons) plotted as “decorrelation” for the data represented in (c) [Color figure can be viewed at wileyonlinelibrary.com]

Figure 11

Heterogeneous afferent connectivity remains the dominant form of heterogeneity towards achieving channel decorrelation despite increase in the number of afferent inputs from EC. (a) Firing rate maps of five different GCs in a network made of a heterogeneous population of 500 GCs and 75 BCs, shown for cases when the network's external inputs were identical (top row) and heterogeneous (bottom row). (b) Cumulative distribution of response correlation coefficients represented for identical (left) and heterogeneous (right) external inputs. (c) Statistical (mean ± SEM) comparison of correlation coefficients obtained with networks endowed with distinct forms of heterogeneities, receiving identical (solid boxes; derived from panel b, left) versus heterogeneous (open boxes; derived from panel b, right) external inputs. (d) Response (output) correlation plotted as a function of input correlation for identical and heterogeneous external inputs [Color figure can be viewed at wileyonlinelibrary.com]

Figure 12

Heterogeneous afferent connectivity remains the dominant form of heterogeneity toward achieving channel decorrelation in a small DG network. (a) Cumulative distribution of correlation coefficients for firing rates of 100 granule cells, computed when the simulations were performed with different sets of fixed values of P _AMPAR (spanning 0.007–20 μm/s) and P _GABAAR (spanning 7–300 nm/s). These simulations were performed in networks constructed with heterogeneous populations of 100 GCs and 15 BCs, with fixed synaptic strengths. (b) Cumulative distribution of pair‐wise correlation coefficients computed from granule cell firing rates in networks constructed with different forms of heterogeneities. 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. (e) Statistical (mean ± SEM) comparison of correlation coefficients obtained with networks endowed with distinct forms of heterogeneities, receiving identical (solid boxes; derived from panel d, top) versus heterogeneous (open boxes; derived from panel d, bottom) external inputs. (f) Response (output) correlation plotted as a function of input correlation for identical and heterogeneous external inputs [Color figure can be viewed at wileyonlinelibrary.com]

Figure 13

Channel decorrelation in a network receiving heterogeneous external input as a function of neuronal diameter and dependence of input–output correlation on the specific kernel used to compute instantaneous firing rate. (a) Cumulative distribution of correlation coefficients of firing rates computed from granule cell firing rates in networks comprised of 100 GCs and 15 BCs, constructed with different forms of age‐related heterogeneities: fully immature, fully mature, neurogenesis‐induced structural heterogeneity of both GC and BC, and neurogenesis‐induced structural heterogeneity only in GC. Panels on the left and right respectively correspond to networks receiving identical and heterogeneous external inputs from the EC. All four populations were endowed with intrinsic and synaptic heterogeneities. (b) Firing rates, represented as quartiles, of all the GCs plotted for the different networks (heterogeneous input vs identical input case) they resided in. (c) Statistical (mean ± SEM) comparison of correlation coefficients obtained with networks endowed with distinct forms of heterogeneities, receiving identical (solid boxes; derived from Panel a, left) versus variable (open boxes; derived from Panel a, right) external inputs. (d) Response (output) correlation plotted as a function of input correlation for identical and heterogeneous external inputs. (e–g) Response (output) correlation plotted as a function of input correlation. Shown are three different plots with the firing rate response correlations computed with different values of σ _FR, the standard deviation of the Gaussian kernel used to convert spike trains to instantaneous firing rates (Supporting Information, Figure S1) [Color figure can be viewed at wileyonlinelibrary.com]