Charge Deficiency, Charge Transport and Comparison of Dimensions

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.Comment: 23 pages, no figure

Consequences of converting graded to action potentials upon neural information coding and energy efficiency

Information is encoded in neural circuits using both graded and action potentials, converting between them within single neurons and successive processing layers. This conversion is accompanied by information loss and a drop in energy efficiency. We investigate the biophysical causes of this loss of information and efficiency by comparing spiking neuron models, containing stochastic voltage-gated Na+ and K+ channels, with generator potential and graded potential models lacking voltage-gated Na+ channels. We identify three causes of information loss in the generator potential that are the by-product of action potential generation: (1) the voltage-gated Na+ channels necessary for action potential generation increase intrinsic noise and (2) introduce non-linearities, and (3) the finite duration of the action potential creates a ‘footprint’ in the generator potential that obscures incoming signals. These three processes reduce information rates by ~50% in generator potentials, to ~3 times that of spike trains. Both generator potentials and graded potentials consume almost an order of magnitude less energy per second than spike trains. Because of the lower information rates of generator potentials they are substantially less energy efficient than graded potentials. However, both are an order of magnitude more efficient than spike trains due to the higher energy costs and low information content of spikes, emphasizing that there is a two-fold cost of converting analogue to digital; information loss and cost inflation

The information encoded in the pseudo-generator potentials of the spiking neuron model.

<p>A. Action potentials (top black trace) evoked by white noise current stimuli (bottom red trace). Upper grey trace: The same voltage response with the action potentials removed and replaced with a linear interpolation of the voltage. This is the pseudo-generator potential, which is an approximation of the generator potential. Lower blue trace: A voltage response of the graded neuron model to the current stimulus shown in the bottom trace. B. The PDF of the pseudo-generator voltage response. (Inset) A QQ plot showing departures from a Gaussian distribution (dotted red-line) for the time-series shown in A (upper grey trace). C. Information rates (bits s⁻¹) of pseudo-generator potentials evoked by white noise current stimuli with different means and standard deviations. The stimuli are identical to those in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003439#pcbi-1003439-g002" target="_blank">Figures 2</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003439#pcbi-1003439-g003" target="_blank">3</a>.</p

The effect of stimulus statistics upon the rate, timing and precision of action potentials.

<p>A. Firing rates (spikes s⁻¹) of the spiking neuron model evoked by white noise current stimuli with different means and standard deviations. The current stimuli used in A–D were identical to those in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003439#pcbi-1003439-g003" target="_blank">Figure 3A</a>. B. Total entropy (bits s⁻¹), C. Noise entropy (bits s⁻¹), and D. Information rate per spike (bits spike⁻¹) of the spiking neuron model.</p

The information rates of the spiking neuron model are robust to voltage-gated ion channel noise.

<p>A. The firing rates of the spiking neuron model (stochastic voltage-gated Na⁺ and K⁺ channels), a modified model with stochastic voltage-gated Na⁺ and deterministic voltage-gated K⁺ channels and a modified model with deterministic voltage-gated Na⁺ and stochastic voltage-gated K⁺ channels evoked by low mean, high standard deviation or high mean, low standard deviation input stimuli. B. The total entropy, C. The noise entropy, and D. The mutual information rates of the same models shown in A evoked by the same stimuli.</p

The energy efficiency of information encoding in spike trains, pseudo-generator potentials and graded potentials.

<p>A. Energy efficiency of information processing (bits ATP molecule⁻¹) with (thick lines; SNR = 2) and without (thin lines) extrinsic noise in the spiking neuron model and the model containing the pseudo-generator potentials, and B. the graded potential model evoked by white noise current stimuli with different means and standard deviations.</p

The effects of channel noise upon sub-threshold and graded voltage signals.

<p>A. The standard deviation of the voltage of the spiking neuron model (stochastic voltage-gated Na⁺ and K⁺ channels), a modified model with stochastic voltage-gated Na⁺ and deterministic voltage-gated K⁺ channels, a modified model with deterministic voltage-gated Na⁺ and stochastic voltage-gated K⁺ channels, and the graded neuron model (stochastic voltage-gated K⁺ channels) over a 16 mV range of holding potentials. B. Shannon information rates of all four models shown in A evoked by low mean, high standard deviation current stimuli at sub-threshold holding potentials. C. Coherence-based information rates of all four models shown in A evoked by low mean, high standard deviation current stimuli at sub-threshold holding potentials. D. Normalized mean square error (nRMSE) information rates of all four models shown in A evoked by low mean, high standard deviation current stimuli at sub-threshold holding potentials.</p

Information encoding in the spiking and graded neuron models.

<p>A. Information rates (bits s⁻¹) of the spiking neuron model evoked by white noise current stimuli with different means and standard deviations. B. Information rates (bits s⁻¹) of the graded neuron model evoked by the same white noise current stimuli as in A.</p

Voltage responses of spiking and graded potentials.

<p>A. The band-limited 300 Hz filtered Gaussian white noise current stimulus. B. The probably density function (PDF) of the current stimulus shown in A. C. A voltage response of the spiking neuron model to the current stimulus shown in A. D. The PDF of the spiking neuron model's voltage response. E. A voltage response of the graded neuron model to the current stimulus shown in A. F. The PDF of the graded neuron model's voltage response. (Inset) A QQ plot showing departures from a Gaussian distribution (dotted red-line) for the time-series shown in E.</p