The Computational Properties of a Simplified Cortical Column Model

<div><p>The mammalian neocortex has a repetitious, laminar structure and performs functions integral to higher cognitive processes, including sensory perception, memory, and coordinated motor output. What computations does this circuitry subserve that link these unique structural elements to their function? Potjans and Diesmann (2014) parameterized a four-layer, two cell type (i.e. excitatory and inhibitory) model of a cortical column with homogeneous populations and cell type dependent connection probabilities. We implement a version of their model using a displacement integro-partial differential equation (DiPDE) population density model. This approach, exact in the limit of large homogeneous populations, provides a fast numerical method to solve equations describing the full probability density distribution of neuronal membrane potentials. It lends itself to quickly analyzing the mean response properties of population-scale firing rate dynamics. We use this strategy to examine the input-output relationship of the Potjans and Diesmann cortical column model to understand its computational properties. When inputs are constrained to jointly and equally target excitatory and inhibitory neurons, we find a large linear regime where the effect of a multi-layer input signal can be reduced to a linear combination of component signals. One of these, a simple subtractive operation, can act as an error signal passed between hierarchical processing stages.</p></div

Nonlinearity resulting from drive into L5 inhibitory population.

<p>This expanded view of row 3, right column of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005045#pcbi.1005045.g005" target="_blank">Fig 5</a> demonstrates the strongly inhomogeneous response resulting from driving the L5 inhibitory population. Solid and dashed lines indicate perturbations of excitatory and inhibitory populations, respectively (L2/3, blue; L4, black; L5, red; L6, green). Simulations in NEST (100 averaged leaky integrate-and-fire simulations) are plotted as points.</p

Homogeneity of output perturbations.

<p>As the magnitude of the input drive increases, the output perturbation responds either linearly (balanced target specificity) or nonlinearly, demonstrating that the target specificity of the incoming input shapes the linearity of the response. Solid and dashed lines indicate perturbations of excitatory and inhibitory populations, respectively (L2/3, blue; L4, black; L5, red; L6, green). For example, when L5 (i.e. 3rd row) is driven with balanced target specificity (middle column), the perturbations increase proportionally as input strength increases from 0 to 20 Hz. In contrast, when the inhibitory population is selectively driven (right column), the perturbation increases sub-linearly as input strength increases. Also included in the third row (dots) are results from 100 averaged leaky integrate-and-fire simulations (performed in NEST, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005045#sec002" target="_blank">Methods</a>), demonstrating that the trends observed in the population density simulations are present in the original system as well. To quantify linearity, the perturbation resulting from a 5 Hz amplitude step input is extrapolated to 10 Hz, and the percentage deviation of this prediction from the true perturbation relative to pre stimulus steady-state is computed. This relative error (See Main Text, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005045#pcbi.1005045.e022" target="_blank">Eq 16</a>) is then normalized by the relative error from the balanced case, demonstrating that in general, balanced inputs result in more linear responses. The table on the right summarizes the relative errors for the different conditions.</p

Additivity of output perturbations.

<p>All possible pairs of two input layers are selected to receive a 10 Hz step input in excess of background excitation. Dots and crosses indicate perturbations of excitatory (top) and inhibitory (bottom) populations computed by DiPDE, respectively (L2/3, blue; L4, black; L5, red; L6, green). For each possible pair of these input layers, the output of all populations is plotted versus the linear superposition of the same inputs taken independently, in two separate simulations (see main text for more details). Linear additivity of inputs between input layers results in points along the dashed identity line; this is the case for excitatory and balanced target specificity. As quantified by the coefficient of determination (<i>r</i>²) resulting from a linear fit, the balanced input regime results in a linear response, and deviates weakly from linearity when inputs are not balanced (See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005045#pcbi.1005045.s002" target="_blank">S2 Fig</a>).</p

Summary of output perturbations evoked by single-layer step inputs.

<p>Step inputs are projected into a drive layer (L2/3, L4, L5, L6) under 3 different target specificity conditions. In all cases a 20 Hz firing rate input in excess of background excitation is applied at 100 ms, and output perturbation is defined as the difference between the eventual perturbed steady-state (200 ms) and the steady-state before perturbation. For example, balanced L2/3 drive (yellow, top-left panel) evokes an approximately 1.5 Hz decrease in firing rate in the L5e subpopulation, while balanced L4 drive (yellow, top-right panel) evokes a 1.5 Hz increase.</p

Model overview and comparison of population statistic model (DiPDE) with leaky integrate-and-fire (LIF) simulations.

<p>(a) Schematic overview of the connections between source (left) and target (right) populations. Line thickness corresponds to the number of projections between populations. The number of projections from inhibitory populations is scaled by 4 (the relative difference between inhibitory and excitatory synaptic strengths), so that excitatory and inhibitory projections can be visually compared. (b) The column model is perturbed in one of three ways: Excitatory, Balanced, or Inhibitory, illustrated here driving Layer 4. (c-d) Mean firing rate across all populations (1 ms bin width) of 100 averaged LIF simulations (solid fluctuating red trace for excitatory and blue for inhibitory subpopulation; a single example firing rate trace is rendered semi-transparent in the background) of the cortical column model. Black lines show prediction of DiPDE simulation across all layers under either (c) 20 Hz step or (d) sinusoidal with 20 Hz peak amplitude in firing rate, 24 Hz frequency inputs in excess of background excitation. At <i>t</i> = 100 ms, an additional input beyond the background excitation drives the excitatory population of layer 4 as illustrated in (b), “Excitatory”. Transient dynamics, steady-state firing rates, and responses to additional inputs are well approximated by DiPDE.</p

Compound synthesis and spectral characterisation from <i>In situ</i> epoxide generation by dimethyldioxirane oxidation and the use of epichlorohydrin in the flow synthesis of a library of β-amino alcohols

Compound synthesis, optimisation and spectral characterisatio

Computing extracellular potential.

<p>(<b>A</b>) Schematic of the compartmental model of a cell in relationship to the recording electrode. The calculation of the extracellular potential involves computing the transfer resistances <i>R_mn</i> between each n-th dendritic segment and m-th recording site on the electrode. (<b>B</b>) Extracellular spike “signatures” of individual cells recorded on the mesh electrode (black dots), using two single-cell models from the layer 4 network model as examples: PV2 (left) and Nr5a1 (right). (<b>C</b>) Modeled extracellular recordings with the linear electrode positioned along the axis of the cylinder in the layer 4 model (left). Extracellular potential responses (right) show all simulated data (color map) as well as from six select channels (black traces superimposed on the color map).</p

Computational performance.

<p>(<b>A</b>) Scaling of wall time duration (normalized by the duration on a single CPU core) with the number of CPU cores for the simulation set up (blue circles) and run (red circles) of the layer 4 model (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0201630#pone.0201630.g005" target="_blank">Fig 5</a>). The ideal scaling is indicated by the dashed line. (<b>B</b>) Wall time increase when computing the extracellular potential for both set up (blue circles) and run (red circles) durations. (<b>C</b>) Scaling of the wall time with the simulated time for a long simulation. The non-ideal scaling with the increase in the number of cores corresponds to the deviations from the dashed line in (A).</p