Effective stimuli for constructing reliable neuron models

Abstract

The rich dynamical nature of neurons poses major conceptual and technical challenges for unraveling their nonlinear membrane properties. Traditionally, various current waveforms have been injected at the soma to probe neuron dynamics, but the rationale for selecting specific stimuli has never been rigorously justified. The present experimental and theoretical study proposes a novel framework, inspired by learning theory, for objectively selecting the stimuli that best unravel the neuron's dynamics. The efficacy of stimuli is assessed in terms of their ability to constrain the parameter space of biophysically detailed conductance-based models that faithfully replicate the neuron's dynamics as attested by their ability to generalize well to the neuron's response to novel experimental stimuli. We used this framework to evaluate a variety of stimuli in different types of cortical neurons, ages and animals. Despite their simplicity, a set of stimuli consisting of step and ramp current pulses outperforms synaptic-like noisy stimuli in revealing the dynamics of these neurons. The general framework that we propose paves a new way for defining, evaluating and standardizing effective electrical probing of neurons and will thus lay the foundation for a much deeper understanding of the electrical nature of these highly sophisticated and non-linear devices and of the neuronal networks that they compose.

Figure 1. From experimental data to acceptable conductance-based neuron model.

(a) Data is collected from voltage responses to a set of repeated intracellular current injections (steps, ramps, noise currents) recorded from single cells' somata. Two repetitions of a step current injection are shown. Two traces with fairly large differences were chosen to highlight the variability. (b) The voltage traces are characterized using a set of features (e.g. firing rate, height of action potentials). For each feature both the experimental mean and standard deviation (SD) are obtained from 15 repetitions of the same stimulus. (c) The generic form of a model to be constrained consists of a reconstructed morphology and an assumed set of membrane ion channels (including their kinetics but not their densities, g_i). (d) A multiple objective, genetic algorithm-based process of stochastic optimization is applied in order to obtain values for g_i that minimize the distance between the experimentally measured set of features and those of the model. The convergence of the average error is shown by the blue curves, one curve for each of three independent applications of the model constraining procedure (e) For the many possible solutions at the final iteration, a selection criterion of two experimental SDs in each feature is used for choosing a subset of solutions (sets of g_i values); these are considered acceptable models. Shown are two out of the six features considered for step stimuli (see Materials and methods) (f) An example of the response of two different successful models to a step current input as in a. Two models with fairly large differences were chosen to highlight the variability. The reconstructed L5 pyramidal cell shown in c is used throughout Figures 1– 6.

Figure 2. Training and Generalization paradigm.

Example - training on responses to step currents and generalizing to responses for ramp currents. (a) Experimental voltage responses recorded from rat layer 5 pyramidal cell (depicted in Figure S1c) to three depolarizing current steps (lower blue) are used as the training set; the experimental response to the largest step current, #3, is displayed in black. (b) Model response to step current #3 following training on the three current steps. (c) Model response (red trace) to a new stimulus, in this case a ramp current (lower red trace in d). (d) Experimental response to the same ramp current. Comparison between model prediction and experimental data, using feature-based distance functions, enables one to quantify the accuracy of the generalization procedure (see Figure 3). In this case the average feature error was approximately 1.5 in units of the experimental standard deviation.

Figure 3. Asymmetric generalization for step and ramp current stimuli.

(a) Models were trained on step currents and generalization tested on ramp currents. Mean and standard deviation of training error (blue) and generalization (red) for increasing number of stimuli included in the training set. (b) Models were trained on ramp stimuli and tested for generalization on step stimuli. (c) Space of acceptable solutions for two out of eight ion channel conductances used in this study. Transient sodium (gNa) and fast potassium (gKv3.1) conductances are shown for models trained on one current step (region in light blue) and models trained on four currents steps (dark blue). (d) Space of acceptable solutions for both step and ramp stimuli (four stimuli in each case) for the two ion channels depicted in (c). The intersection area (darker blue) represents solutions that are consistent with both stimuli types.

Figure 4. Within stimulus generalization.

(a) Models were trained on ramp stimuli and tested for generalization on ramp stimuli Mean and standard deviation of training error (blue) and generalization (red) for increasing number of stimuli included in the training set. (b) Models were trained on step currents and generalization tested on different intensities of step currents.

Figure 5. Generalization based on step+ramp stimuli outperforms generalization based on noisy stimuli.

(a) Models were trained on a combined set of step and ramp stimuli (schematics at left, blue) and tested for generalization on noisy inputs. Experimental response (black) is displayed along with one model response (red). Timing of spikes is highlighted by corresponding color dots at top. GCF value 0.92 (b) Generalization of models, trained using the type 1 noisy stimulus, to the type 2 noisy stimulus (black – experimental response; red – model response to type 2 noisy input). GCF value 0.93 (c) Generalization of models trained using the type 1 noisy stimulus to step current pulses (black – experimental response, red – response to type 2 noisy input). (d) Generalization of models trained on type 1 noisy input to ramp current pulses. Note considerable mismatch in both c and d.

Figure 6. Constraining conductance-based models for different neuron types.

(a) Experimental response (black traces) to a 2 second long step current pulse (lower trace, black) and model response (blue traces) to the same current pulse; training set consisted of the combined step and ramp currents. (b) Generalization was tested on the high mean, low variability type 1 noisy current pulse (bottom grey). Experimental response (black) and one model response (red) are shown along with corresponding color dots indicating timing of APs. GCF values: 0.91, 0.89, 0.92 top to bottom respectively (c) Generalization error (red) and training error (blue) for models trained on step currents and generalization tested on ramp currents. Cell 2 - L5 pyramidal cell from a juvenile rat (p16); cell 3 - fast-spiking interneuron, juvenile rat (p16); Cell 4 - pyramidal cell from a young mouse (p34). Corresponding morphology is shown at left.