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Conceived and designed the experiments: DBF DP. Performed the experiments: JRC. Analyzed the data: DBF. Wrote the paper: DBF DP JRC.

The authors have declared that no competing interests exist.

An important problem in neuronal computation is to discern how features of stimuli control the timing of action potentials. One aspect of this problem is to determine how an action potential, or spike, can be elicited with the least energy cost, e.g., a minimal amount of applied current. Here we show in the Hodgkin & Huxley model of the action potential and in experiments on squid giant axons that: 1) spike generation in a neuron can be highly discriminatory for stimulus shape and 2) the optimal stimulus shape is dependent upon inputs to the neuron. We show how polarity and time course of post-synaptic currents determine which of these optimal stimulus shapes best excites the neuron. These results are obtained mathematically using the calculus of variations and experimentally using a stochastic search methodology. Our findings reveal a surprising complexity of computation at the single cell level that may be relevant for understanding optimization of signaling in neurons and neuronal networks.

Computational neuroscience seeks to understand the mechanisms by which signals excite a neuron or a neuronal network. An important consideration in these studies is optimality, i.e., what signal most effectively causes excitation. Optimization of neuronal signaling is important for networks that need to minimize energy costs, for sensory neurons to selectively respond to specific stimulus features, and for therapeutic deep brain stimulators to maximize battery life. Here we show in a classic mathematical model of the action potential and in experiments on a single cell preparation that: 1) a single neuron can be highly discriminatory for the shape of low amplitude stimuli that elicit an action potential and 2) the shape of the optimal stimulus depends upon the overall state of inputs to the neuron. Our findings reveal a surprising complexity of computation at the single cell level that may be important for understanding physiological function of the nervous system.

A central question in neuronal computation is to determine the features of neural stimuli that cause action potentials

One method for determining optimal signals is the calculus of variations

An important step in addressing these questions is the development of a theory of optimality in single neurons. This theory should account for the complex, multi-scale and nonlinear behavior of a neuron. For example, several mechanisms are known to generate an action potential including membrane depolarization and post-inhibitory rebound excitation

Stimulation trajectories _{1} induce a state change from rest α_{0} to threshold β_{1} by means of depolarization, and stimulation curves _{2} induce a state change to β_{2} by means of post-inhibitory rebound. Red trajectories illustrate the optimal paths for which total current is minimized; blue trajectories are neighboring paths of suboptimal stimulation.

The signals a neuron receives are combinations of post-synaptic currents (PSCs), which can be either excitatory or inhibitory. The duration of PSCs can vary considerably depending on cell type

In the present study we investigate stimulus optimization principles using one of the best characterized experimental preparations - the squid giant axon - and its mathematical representation, the Hodgkin & Huxley model and a recent modification of the model

The Hodgkin & Huxley model _{Na}_{K}_{K}_{stim}_{0} to threshold β_{1} along the trajectory in _{stim}_{0} to β_{1} with a minimum amount of applied root mean square (RMS) current (red tracing in _{0} in _{Na}_{2} in _{2} at the end of the hyperpolarizing pulse is below the resting level (_{0} and β_{2} is illustrated in _{0} and β_{2}) that minimized the amount of current required for the anode break result. The

_{0} to threshold β_{1} and post-inhibitory rebound stimulation induces excitation via state change from rest α_{0} to threshold β_{2}. Blue trajectories are paths corresponding to rectangular current pulses; red trajectories are the optimized paths computed from the model with the calculus of variations. Note the small loop (

In the above analysis the only restrictions placed on the current _{stim}_{0} to β_{1} (or β_{2}) in 20 msec with minimal RMS current. This approach is relevant for exogenous stimulation of a neuron that occurs, for example, during deep brain stimulation _{stim}_{stim}_{syn}_{syn}_{syn}_{syn}_{syn}_{stim}

As noted above, stochastic perturbations can also be used to determine stimulus optimization without requiring a mathematical description of the underlying dynamics

These results are all from a single axon. Similar results were observed in all other axons. ^{−1}. Each PSC had a decay constant of 1 msec. The stimulation amplitude (RMS) was set to a level that produced rare action potentials (<1 Hz).

Next we asked how close any 20 msec portion of the input signal (

The experimental protocol and analysis illustrated in

Each signal was normalized to its maximum value that occurred within a few msec before a spike.

A visual comparison of the noise-derived optimal signal in _{stim}

The results of the analysis described above are illustrated in

We note that the theoretically derived waveforms in

The analysis of

This experiment was carried out as in

As noted above (

To further explore the difference between short and long PSCs we increased the excitability of the axon, as demonstrated previously

We have shown that a single neuron can be highly discriminatory for the shape of low amplitude stimuli that elicit an action potential and that the shape of the optimal stimulus is dependent upon input context, i.e., the optimal stimulus for eliciting a spike is determined by the nature and the type of all inputs to the neuron. Our results validate two methodologies to study optimality in neuronal systems. Using the calculus of variations, we determined optimal signals for the Hodgkin & Huxley model. This theory predicts that our stochastic search methodology derived from experiments should converge to the optimal stimulus derived from the theoretical approach, a prediction that is supported by the results in

Our results indicate that questions of optimality are more complex than the one model-one optimal view that is widely found in the discussions of neuronal excitation. While simpler qualitative models which are more amenable to mathematical analysis than ionic models can also be used to qualitatively predict optimal signal, they may miss the multiple locally optimal signals that are needed to understand the full landscape of neuronal signaling. For example, an integrate-and-fire model does not predict post-inhibitory rebound excitation nor does it predict neuronal firing with inputs consisting solely of inhibitory PSCs. Multiple optimal signals could allow a neuron to be responsive to a wider range of stimuli, where stimulus context is key to understanding neuronal optimality. As further details of this context are considered

We have shown that PSC duration is an important factor in stimulus optimization (

The Hodgkin & Huxley model is given by^{2}), _{Na}_{K}_{L}^{+}, K^{+}, and leak current, respectively, with _{Na}_{K}_{L}_{stim}^{2}. The voltage- and time-dependent variables _{m}(_{m}(_{h}(_{h}(_{n}(_{n}(^{−1}. These equations were taken directly from Hodgkin & Huxley _{n} = 0.125 φexp(-_{n} = 0.125 φexp(-

We minimize the L^{2} norm of the applied current _{stim}_{V}_{m}_{n}_{h}_{off}_{V}_{m}_{n}_{h}_{0} in _{V}_{m}_{n}_{h}_{1} in _{0}, to β_{1}. Similar results were found with Mathematica (Wolfram Research; Champaign, Il) using a shooting method. For post-inhibitory rebound (PIR) stimuli, we repeated the above methods with a hyperpolarizing pulse 10 msec in duration. As noted above (

Let

Experiments were carried out on squid giant axons using methods previously described

_{stim}_{stim}_{syn}

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_{syn}

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Application of the calculus of variations to endogenous stimulation.

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We thank Xuanxuan Gan for programming support.