Correlation-based model of artificially induced plasticity in motor cortex by a bidirectional brain-computer interface

Experiments show that spike-triggered stimulation performed with Bidirectional Brain-Computer-Interfaces (BBCI) can artificially strengthen connections between separate neural sites in motor cortex (MC). What are the neuronal mechanisms responsible for these changes and how does targeted stimulation by a BBCI shape population-level synaptic connectivity? The present work describes a recurrent neural network model with probabilistic spiking mechanisms and plastic synapses capable of capturing both neural and synaptic activity statistics relevant to BBCI conditioning protocols. When spikes from a neuron recorded at one MC site trigger stimuli at a second target site after a fixed delay, the connections between sites are strengthened for spike-stimulus delays consistent with experimentally derived spike time dependent plasticity (STDP) rules. However, the relationship between STDP mechanisms at the level of networks, and their modification with neural implants remains poorly understood. Using our model, we successfully reproduces key experimental results and use analytical derivations, along with novel experimental data. We then derive optimal operational regimes for BBCIs, and formulate predictions concerning the efficacy of spike-triggered stimulation in different regimes of cortical activity.Comment: 35 pages, 9 figure

From Squid to Mammals with the HH Model through the Nav Channels’ Half-Activation-Voltage Parameter

The model family analyzed in this work stems from the classical Hodgkin-Huxley model (HHM). for a single-compartment (space-clamp) and continuous variation of the voltage-gated sodium channels (Na-v) half-activation-voltage parameter Delta V-1/2, which controls the window of sodium-influx currents. Unlike the baseline HHM, its parametric extension exhibits a richer multitude of dynamic regimes, such as multiple fixed points (FP's), bi- and multistability (coexistence of FP's and/or periodic orbits). Such diversity correlates with a number of functional properties of excitable neural tissue, such as the capacity or not to evoke an action potential (AP) from the resting state, by applying a minimal absolute rheobase current amplitude. The utility of the HHM rooted in the giant squid for the descriptions of the mammalian nervous system is of topical interest. We conclude that the model's fundamental principles are still valid (up to using appropriate parameter values) for warmer-blooded species, without a pressing need for a substantial revision of the mathematical formulation. We demonstrate clearly that the continuous variation of the Delta V-1/2 parameter comes close to being equivalent with recent HHM 'optimizations'. The neural dynamics phenomena described here are nontrivial. The model family analyzed in this work contains the classical HHM as a special case. The validity and applicability of the HHM to mammalian neurons can be achieved by picking the appropriate Delta V-1/2 parameter in a significantly broad range of values. For such large variations, in contrast to the classical HHM, the h and n gates' dynamics may be uncoupled - i.e. the n gates may no longer be considered in mere linear correspondence to the h gates. Delta V-1/2 variation leads to a multitude of dynamic regimes-e.g models with either 1 fixed point (FP) or with 3 FP's. These may also coexist with stable and/or unstable periodic orbits. Hence, depending on the initial conditions, the system may behave as either purely excitable or as an oscillator. Delta V-1/2 variation leads to significant changes in the metabolic efficiency of an action potential (AP). Lower Delta V-1/2 values yield a larger range of AP response frequencies, and hence provide for more flexible neural coding. Such lower values also contribute to faster AP conduction velocities along neural fibers of otherwise comparable-diameter. The 3 FP case brings about an absolute rheobase current. In comparison in the classical HHM the rheobase current is only relative - i.e. excitability is lost after a finite amount of elapsed stimulation time. Lower Delta V-1/2 values translate in lower threshold currents from the resting state

Energy-Optimal Electrical-Stimulation Pulses Shaped by the Least-Action Principle

Electrical stimulation (ES) devices interact with excitable neural tissue toward eliciting action potentials (AP's) by specific current patterns. Low-energy ES prevents tissue damage and loss of specificity. Hence to identify optimal stimulation-current waveforms is a relevant problem, whose solution may have significant impact on the related medical (e. g. minimized side-effects) and engineering (e. g. maximized battery-life) efficiency. This has typically been addressed by simulation (of a given excitable-tissue model) and iterative numerical optimization with hard discontinuous constraints - e.g. AP's are all-or-none phenomena. Such approach is computationally expensive, while the solution is uncertain - e. g. may converge to local-only energy-minima and be model-specific. We exploit the Least-Action Principle (LAP). First, we derive in closed form the general template of the membrane-potential's temporal trajectory, which minimizes the ES energy integral over time and over any space-clamp ionic current model. From the given model we then obtain the specific energy-efficient current waveform, which is demonstrated to be globally optimal. The solution is model-independent by construction. We illustrate the approach by a broad set of example situations with some of the most popular ionic current models from the literature. The proposed approach may result in the significant improvement of solution efficiency: cumbersome and uncertain iteration is replaced by a single quadrature of a system of ordinary differential equations. The approach is further validated by enabling a general comparison to the conventional simulation and optimization results from the literature, including one of our own, based on finite-horizon optimal control. Applying the LAP also resulted in a number of general ES optimality principles. One such succinct observation is that ES with long pulse durations is much more sensitive to the pulse's shape whereas a rectangular pulse is most frequently optimal for short pulse durations

The <i>V</i>_1/2 parameter values [mV], by ion-channel type—from [24, 25].

<p>The <i>V</i>_1/2 parameter values [mV], by ion-channel type—from [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143570#pone.0143570.ref024" target="_blank">24</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143570#pone.0143570.ref025" target="_blank">25</a>].</p

Bifurcation-terms Glossary [43, 49].

<p>Bifurcation-terms Glossary [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143570#pone.0143570.ref043" target="_blank">43</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143570#pone.0143570.ref049" target="_blank">49</a>].</p

A generalization perspective of Δ<i>V</i>_1/2 variation effects—from a compartment to a cell.

<p><b>A</b> When <i>I</i>_<i>STIM</i> is lower, it latently reaches and depolarizes compartments down the cable. This is clearly visible from the voltage traces, which are depolarized up to halfway down. As a result, when the AP is finally evoked it propagates faster. <b>B</b> “supra-threshold” case with a safety factor applied <math><mrow><msub><mi>I</mi><mrow><mi>S</mi><mi>T</mi><mi>I</mi><mi>M</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msub><mi>I</mi><mo>^</mo><mrow><mi>T</mi><mi>H</mi><mi>R</mi></mrow></msub></mrow></math>.</p

State-space dynamics and model regimes as a function of Δ<i>V</i>_1/2.

<p>These bifurcation diagrams illustrate the FP’s and PO’s of the <i>V</i> and <i>h</i> model states as a function of Δ<i>V</i>_1/2. <i>Thin black line</i>: stable FP’s; <i>thick cyan</i>: the unstable-FP branches (with one or two (real or complex) positive-real eigenvalues. <i>LP</i>1 and <i>LP</i>2: lower and upper limits of the middle branch. HB: Hopf bifurcation. See the text for more details. Thin and thick dashed lines: minimum and maximum model-state value for the stable and unstable cycles respectively. Unstable cycles are born from FP’s at a <i>subcritical</i> HB and metamorphose to stable at a cycle saddle-node (fold).</p

Resting threshold current <i>I</i>_<i>THR</i> and mid-axon mean AP conduction velocity.

<p>Notice the clear linear trend of <i>I</i>_<i>THR</i> increasing with Δ<i>V</i>_1/2 for both <i>T</i>_<i>STIM</i> cases. A significant trend of accelerating AP propagation with Δ<i>V</i>_1/2 can be acknowledged. To prevent excessive results variability (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143570#pone.0143570.g008" target="_blank">Fig 8</a> and text), conduction velocity was computed for <i>I</i>_<i>STIM</i> = 1.5 × <i>I</i>_<i>THR</i>.</p

Gate-state dynamics parameters (see also Box 1—key notes on gate-state dynamics).

<p><i>Notes:</i>:</p><p><b>*1:</b> The temperature-dependence factor. Please see the first note in Box 1.</p><p><b>*2:</b> Gate-state dynamics parameters of the <i>K</i>⁺ or <math><mrow><mi>N</mi><msubsup><mi>a</mi><mrow><mi>v</mi><mn>1</mn><mo>.</mo><mi>X</mi></mrow><mo>+</mo></msubsup></mrow></math> channels. Actual channel opening/closing rates are given by Boltzmann-like functions—for details see the second note in Box 1.</p><p><b>*3:</b> The <math><mrow><mi>N</mi><msubsup><mi>a</mi><mrow><mi>v</mi><mn>1</mn><mo>.</mo><mi>X</mi></mrow><mo>+</mo></msubsup></mrow></math> channels’ inactivating <i>h</i> gates have an <i>asymptotic</i> state, which is independent of the <i>h</i> gates’ opening/closing rates—see the second and third notes in Box 1.</p><p>Gate-state dynamics parameters (see also Box 1—key notes on gate-state dynamics).</p