Fig 1.
Membrane voltage polarity and stimulation current reference direction.
The membrane voltage v(t) is measured with respect to outside the neuron, in part because the preparation bath is electrically grounded. Stimulation current i(t) is positive charge flow into the neuron under experimental control. A positive i(t) tends to increase the membrane voltage, while a negative i(t) tends to decrease the membrane voltage.
Fig 2.
Library of optimal current stimuli.
Bottom (each frame): Rectangular pulse stimuli i(t) (black) and optimal stimuli i*(t) (red). Top (each frame): Simulated model response r(t) to rectangular pulse stimuli i(t) (black) and response v*(t) to optimal stimuli i*(t) (red). As P and Q are reduced, there is a greater emphasis on energy reduction and tracking of the reference membrane voltage degrades.
Fig 3.
Computed optimal current stimulus and neuron model response for {P = 1, Q= 1, R= 10} and different pulse widths.
The model fails to produce an action potential due to the high level of energy reduction.
Fig 4.
Responses of one leech P-cell to rectangular pulse and optimal current stimuli.
Bottom (each frame): rectangular pulse iexp(t) (black) and optimal stimuli (red) as applied to the neuron (ten traces) including percent energy reduction of optimal stimuli. Top (each frame): Responses of P-cell (ten traces) and correlation coefficient of the average rexp(t) compared to the average
. As the emphasis on energy reduction increases, tracking performance decreases.
Fig 5.
Responses of all neurons to rectangular pulse and optimal current stimuli.
Bottom (each frame): rectangular pulse iexp(t) (black) and optimal stimuli : high-energy case, red {P = 10, Q = 10, R = 1}, medium-energy case, blue {P = 5, Q = 5, R = 1}, and low-energy case, green {P = 1, Q = 1, R = 1}. Ten stimulus response pairs are shown for each {P, Q, R} case. Top (each frame): responses to rectangular pulse and optimal stimuli in matching colors.
Fig 6.
Average and variability of all neuron responses to current stimuli.
Each frame shows the average ± one standard deviation (SD) of fifty total responses (across five animals) computed at each sample time after shifting responses to a resting membrane voltage of 0 mV and shifting action potential peaks to the average latency. Bottom (each frame): representative rectangular pulse iexp(t) (black) and optimal stimulus (red) applied to the neuron. Top (each frame): average response to iexp(t) (black) showing voltages within one standard deviation of the mean shaded in gray. Average response to
(red) showing voltages within one standard deviation of the mean shaded in pink.
Fig 7.
Two optimal current stimuli computed by bvp5c() with different initial mesh sizes.
Two optimal stimuli (top, left) (red) and
(black) obtained from bvp5c() by using an initial mesh size of 125 and 2500 points, respectively. The difference between these two solutions
after interpolation is shown at top right. These two solutions for i*(t), though quantitatively identical, produce significantly different responses in the neuron model Eq (3) as computed by ode45() (bottom, in like colors). The reference signal r(t) is also shown for comparison (blue).
Fig 8.
Sensitivity of neuron model response to ‘blended’ optimal current stimuli.
Responses v*(t) of the neuron model Eq (3) (top) to blended currents Eq (2) (bottom) as computed by ode45(). Small variations in the optimal current create dramatic changes in the membrane voltage response suggesting that this optimal stimulus shape meets minimal conditions for evoking an action potential. Currents that do not produce an action potential and the associated responses are in red.
Fig 9.
State trajectories of the reduced neuron model Eq (3) evoked by the blended currents of Fig 8.
Small changes in the blended current lead to trajectories corresponding to action potentials (black). Those trajectories are separated from trajectories that do not lead to an action potential (red) by a separatrix visible in (d) suggesting that these optimal stimuli satisfy minimal conditions needed to evoke an action potential.
Fig 10.
Simulation and laboratory methods summary.
Block diagram of the Reduced Energy Input Stimulus Discovery Method [27] as applied to stimulation of pressure-sensitive mechanosensory P-cell neurons in the leech. Circled letters are used in the text to reference parts of the diagram. Photo of leech ganglion courtesy T. Groves, Jellies Lab. Waveforms are for illustrative purposes only.
Table 1.
Neuron model parameters [13].