Real dendrites are constrained by current transfer optimality.

<p>(<b>A</b>) Scatter plots of measured radius against optimal radius and correlation coefficients for different cell classes. Different colours denote different cells of the same class. (<b>B</b>) Average current transfer to the root from different cell classes with (from left to right) constant, measured, and optimal radii. (<b>C</b>) Examples of reconstructed sample morphologies and the same morphologies with optimal diameter profiles. The neurons shown in this figure are two types of fly neurons (HS cell and VS cell with specific membrane conductance of <i>g</i>_<i>l</i> = 5 × 10⁻⁴ S/cm²) and three mammalian neurons (dentate gyrus granule cells with <i>g</i>_<i>l</i> = 4 × 10⁻⁵ S/cm², and cerebellar Purkinje cells and neocortical Layer V neurons with <i>g</i>_<i>l</i> = 5 × 10⁻⁵ S/cm²).</p

Cable theory with arbitrary diameters—accuracy of the first-order analytical approximation.

<p>(<b>A</b>) Sample radius profiles illustrating the accuracy of the analytical approximation. The radius profiles are smoothed from top to bottom by increasing the period and reducing the amplitude of the radius change. Current is injected separately at the three points indicated by arrows. (<b>B</b>) Comparison of numerical (red solid lines) and analytical (black dashed lines) voltages in dendrites with varying radius profiles. As the radius changes more slowly, the first-order approximation becomes more accurate (from top to bottom). Note that the large diameters used in this figure emphasize the difference between the numerical and analytical solution. Using smaller diameters as are usual everywhere but at the most proximal dendrites, the analytical approximation becomes essentially a perfect match.</p

Diameter profiles to optimise current transfer.

<p>Comparison of non-parametrically optimised (red solid lines) and theoretical (black dashed lines) radius profiles for different electrotonic lengths of dendritic branch. The theoretical profile tapers more strongly in the shortest case as it neglects the increase in distal input resistance from the sealed end. The scaling parameters <i>α</i> corresponding to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004897#pcbi.1004897.e005" target="_blank">Eq 4</a> are 101.1, 24.08, and 10.81 respectively.</p

Sparsifying (filtering) spike train ensembles leads to increased classical measures of pattern separation, but reduced information content.

A Demonstration of filtering a phase-locked spike train ensemble (left) by random spike deletion (right). The probabilities of spike deletion are p = 0.5, 0.75, 0.85, 0.95. B Standard measures of pattern separation as a function of discretisation bin size on a randomly filtered spike train with p = 0.5. Solid, dashed, and dotted lines refer respectively to strong, medium, and weak input similarities (phase-locked strengths of 0.75, 0.5, and 0.25, see Methods). From darkest to lightest, colours plot standard measures of pattern separation: orthogonalisation Υθ, scaling Υσ, decorrelation Υρ, and Hamming distance Υη. Wasserstein distance Υδ does not rely on discretisation and is not plotted. All values are normalised for comparisons, raw values are plotted in S1(C)–S1(F) Fig. C Standard measures of pattern separation applied to filtered spike trains. Filtering methods and input similarities are as in B. The x-axis gives the filtering parameter in each case. All values are normalised for comparisons, raw values are plotted in S1(G)–S1(K) Fig. D Mutual information between input spike train ensembles and filtered spike train ensembles. Filtering methods and input similarities are as in B. Colours correspond to different neuronal codes (see Methods): from darkest to lightest instantaneous spatial, temporal, local rate, and ensemble rate. All values are normalised for comparisons, raw values are plotted in S2(A)–S2(D) Fig. E Mutual information as a function of classical pattern separation measures. Mutual information was maximised over all spiking codes and bin sizes for different input rates, correlation structures and strengths, and filter types and strengths. Clockwise from top left: orthogonalisation Υθ, decorrelation Υρ, Wasserstein distance Υδ, and Hamming distance Υη. Shaded areas show one standard deviation. Raw scatter plots and densities are shown in S2(E)–S2(H) Fig.</p

S4 Fig -

A Example input spiking rasters (top) and output voltage traces (bottom) for the single granule cell model. B Classical pattern separation measures applied to simulated voltage traces from a granule cell model with varied timescales of synaptic depression. From left to right: orthogonalisation Υθ, scaling Υσ, decorrelation Υρ, Hamming distance Υη, and Wasserstein distance Υδ. Solid lines correspond to a strong input similarity and a binning window (where applicable) of 100 ms, dashed lines correspond to a strong input similarity and a binning window of 10 ms, dash-dotted lines correspond to a weak input similarity and a binning window of 100 ms, and dashed lines correspond to a strong input similarity and a binning window of 10ms. C As above with varied timescales of synaptic facilitation. D As above with varied ion channel spatial heterogeneities. Input spike traces are two minutes long and consist of phase-locked inputs with a phase rate of 0.6Hz and a spiking rate of 5Hz. (TIF)</p

Information theoretic measures can predict pattern completion performance.

A Example binary patterns from the Kuzushiji-49 dataset (top to bottom) with increasing levels of pixel noise (left to right: 0, 200, 400, and 600 pixels). B Top: Completion accuracy of a Hopfield network as a function of noise strength. Middle: Correlations between different classes of pattern as a function of noise strength. Bottom: Relative redundancy reduction (ΥR) between classes of patterns as a function of noise strength. Solid line shows the mean over 1000 repetitions and the shaded area shows the standard error. C Completion accuracy of a Hopfield network against decorrelation Υρ. D Accuracy against ΥR.</p

S1 Fig -

A Example rasters resulting from different methods of generating ensembles. All ensembles have a mean spike rate of 2.5Hz per neuron. Clockwise from top left: Phase-locked ensemble with a phase-locking strength of 0.9 and a phase rate of 0.6Hz, periodic ensemble with α = 10, cross-correlated ensemble with a similarity strength of 0.8, and bursty ensemble with α = 0.5. B Filtering of spike train ensembles. From top to bottom: random filtering with p = 0.5, 0.75, 0.85, 0.95, n-th pass filtering with n = 2, 4, 7, 20, refractory filtering with t = 0.18, 0.54, 1.01, 3.71 s, and competitive filtering with t = 3.4, 13.7, 17.9, 31.4 ms. Filtering parameters are chosen to give roughly equal numbers of spikes across different filters. C Unnormalised orthogonalisation Υθ values for comparison with Fig 1B. D Unnormalised scaling Υσ values for comparison with Fig 1B. E Unnormalised decorrelation Υρ values for comparison with Fig 1B. F Unnormalised Hamming distances Υη values for comparison with Fig 1B. G Unnormalised orthogonalisation Υθ values for comparison with Fig 1C. H Unnormalised scaling Υσ values for comparison with Fig 1C. I Unnormalised decorrelation Υρ values for comparison with Fig 1C. J Unnormalised Hamming distances Υη values for comparison with Fig 1C. K Unnormalised Wasserstein distances Υδ values for comparison with Fig 1C. (TIF)</p

New pattern separation measures can be applied to larger network models.

A Example of inputs to the large network model. Top. Example path in space that generates grid cell-like firing. Colour gradient indicates time over 20s. Bottom: Input firing rasters of 50 neurons over the shown path for 20s. Colours match the time above. B Pattern separation as measured by Hamming distance Υη (top) and sparsity-weighted MI ΥM (bottom) as a function of the maximum firing rate of input cells. Dark lines correspond to 5, 000 inputs to 50, 000 principal cells, and light lines to 10, 000 inputs to 100, 000 principal cells. Error bars show one standard deviation over 24 repetitions. C Pattern separation as a function of the number of principal cells innervated by each of the 100 (50k network) or 200 (100k network) inhibitory basket cells. D Pattern separation as a function of the synaptic strength of the inhibitory basket neurons. A scale of 1 corresponds to 1nS. E Pattern separation as a function of the proportion of principal neurons that are adult-born. Other parameters are as described in Methods and colours and markers are as described in B.</p

Information theoretic measures of pattern separation penalise pattern destruction (information loss).

A Normalised sparsity weighted mutual information ΥM applied to randomly filtered spike train ensembles. Solid, dashed, and dotted lines refer respectively to strong, medium, and weak input similarities (see Methods). Colours correspond to different neuronal codes (see Methods): from darkest to lightest instantaneous spatial, temporal, local rate, and ensemble rate. All values are normalised for comparisons, raw values are plotted in S3(A) Fig. B Normalised sparsity weighted transfer entropy ΥT applied to filtered spike train ensembles. Colours as in panel A. All values are normalised for comparisons, raw values are plotted in S3(B) Fig. C Normalised relative redundancy reduction ΥR applied to filtered spike train ensembles. Colours as in panel A. All values are normalised for comparisons, raw values are plotted in S3(C) Fig.</p

S6 Fig -

A Example dynamic IV curves for mature (dark blue) and adult-born (light blue) granule cells. Circles show the simulated results from full compartmental models and solid lines the best fit under Eq 24. B Example voltage traces for mature granule cells showing spike-frequency adaptation. C Example somatic voltages in a mature granule cell in response to synaptic inputs at increasing distances (lighter blue lines). D Latency of synaptic inputs at the soma (left) and transfer resistance to the soma (right) as a function of relative path length for mature (dark blue) and adult-born (light blue) granule cells. Shaded areas show one standard deviation around the mean. E Adaptive EIF parameter distributions for mature (lower left) and adult-born (upper right) granule cell compartmental models. The leading diagonal panels show single marginal distributions, and the off-diagonal panels show the pairwise marginals. F Examples of grid cell-like firing rates as a function of location (Eq 26). (PDF)</p