J Comput Neurosci DOI 10.1007/s10827-014-0513-9
Context-dependent coding in single neurons Rebecca A. Mease & SangWook Lee & Anna T. Moritz & Randall K. Powers & Marc D. Binder & Adrienne L. Fairhall
Received: 9 October 2012 / Revised: 11 June 2014 / Accepted: 16 June 2014 # Springer Science+Business Media New York 2014
Abstract The linear-nonlinear cascade model (LN model) has proven very useful in representing a neural system’s encoding properties, but has proven less successful in reproducing the firing patterns of individual neurons whose behavior is strongly dependent on prior firing history. While the cell’s behavior can still usefully be considered as feature detection acting on a fluctuating input, some of the coding capacity of the cell is taken up by the increased firing rate due to a constant “driving” direct current (DC) stimulus. Furthermore, both the DC input and the post-spike refractory period generate regular firing, reducing the spike-timing entropy available for encoding time-varying fluctuations. In this paper, we address these issues, focusing on the example of motoneurons in which an afterhyperpolarization (AHP) current plays a dominant role regularizing firing behavior. We explore the accuracy and generalizability of several alternative models for single neurons under changes in DC and variance of the stimulus input. We use a motoneuron simulation to compare coding models in neurons with and without the AHP current. Finally, we quantify the tradeoff between instantaneously encoding information about fluctuations and about the DC. Keywords Spike-triggered Reverse correlation . Covariance modes . Neural filters . Motoneurons . After-hyperpolarization . SK channels Action Editor: Mark van Rossum Electronic supplementary material The online version of this article (doi:10.1007/s10827-014-0513-9) contains supplementary material, which is available to authorized users. R. A. Mease : S. Lee : A. T. Moritz : R. K. Powers : M. D. Binder : A. L. Fairhall (*) University of Washington, Seattle, WA, USA e-mail:
[email protected] R. A. Mease Institute of Neuroscience, Technical University of Munich, Munich, Germany
1 Introduction 1.1 Modeling neuronal responses Reverse correlation (de Boer and Kuyper 1968; Marmarelis and Marmarelis 1968; Rieke et al. 1997; Simoncelli et al. 2004) is a commonly used method to describe the input– output transform of a neural circuit or a single neuron. In this paradigm, the system or neuron is driven with a random, dynamically varying stimulus and the output is measured as a compound field potential or individual neural spike train. After compiling an extended sequence of these responses, one can construct a “linear–nonlinear” (LN) model, in which the firing rate of a neuron or neural system is predicted by linearly filtering the input with the identified feature, and passing the result through a nonlinear function (Korenberg and Hunter 1986; Simoncelli et al. 2004). The linear feature is determined by cross-correlating the input and output to estimate the stimulus feature that causes the system to respond. The nonlinear stage allows one to account for phenomena such as saturation and non-negative firing rates. In applications of these methods to neural systems, the components of the LN model are not easily related to underlying circuit or biophysical properties. When applied to responses of single neurons, one hopes to relate the characteristics of the LN model to ion channel dynamics (Hong et al. 2007; Famulare and Fairhall 2010; Mease et al. 2013). The linear portion of this model is typically found by computing the average stimulus trajectory preceding spikes, or the spike triggered average (STA) (Bryant and Segundo 1976), Fig. 1b. For many neuron types, STAs are characterized by a shallow hyperpolarizing trough followed by a more rapid depolarizing peak immediately preceding the spike (Poliakov et al. 1997; Binder et al. 1999; Aguera y Arcas et al. 2003; Powers et al. 2005). Specific characteristics of the STA have been related to the underlying biophysical
J Comput Neurosci
et al. 2005), and have been used in a handful of cases to elucidate the contributions of specific channel types to shaping the feature selectivity or threshold properties of the neuron (Slee et al. 2005; Hong et al. 2007; Diaz-Quesada and Maravall 2008). While LN models generally assume that each action potential is independent, the likelihood of a neuron to fire an action potential is modulated not only by fluctuating components of the current, but also by its own internal state. The addition of a constant depolarizing ‘drive’ via direct current (DC) injection changes the neuron’s voltage threshold, altering the neuron’s excitability. The neuron’s state also depends on its recent history of activity through refractoriness. More generally, the effects of recent activity can depress (‘spikefrequency adaptation’) or facilitate a neuron’s subsequent firing activity over timescales of hundreds of milliseconds (Koch 1999; Hille 2001; Powers et al. 2005). Both DC inputs and internal dynamics serve to regularize the spike train, profoundly altering the stimulus dependence of the system
mechanisms which govern spiking; for example, the duration of the depolarizing peak in the STA reflects the integration window within which coincident synaptic inputs can be detected (Svirskis et al. 2002). Similarly, the hyperpolarizing trough in the STA may indicate that excitatory inputs are more likely to trigger spikes when excitation is preceded by inhibition (Mainen et al. 1995; Poliakov et al. 1997; Binder et al. 1999; Fellous and Sejnowski 2003; Svirskis et al. 2004). Models that include multiple features (de Ruyter van Steveninck and Bialek 1988; Brenner et al. 2000a, b; Aguera y Arcas et al. 2003; Simoncelli et al. 2004), Fig. 1c, d, usually perform better in predicting output as they capture multiple effects influencing spike firing and allow for more complicated nonlinearities than simple saturation. Such models have been used to reveal complex properties of receptive fields of sensory neurons (Rust et al. 2005; Touryan et al. 2005; Maravall et al. 2007). When applied to single neurons, these techniques allow one to determine the current features that drive individual neurons (Aguera y Arcas et al. 2003; Slee B
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Fig. 1 Example data and analysis. a The membrane potential of an in vitro (top) and simulated (middle) motoneuron during one second of injected noise stimulus (bottom). b The spike-triggered average current preceding each spike time. Time zero is the time of the spike. Onedimensional linear-nonlinear model: c To evaluate the selectivity of the neuron to the STA, individual spike-evoking stimuli were projected onto the STA. Each scalar projection value corresponds to a spike and is the vector dot product of the STA and the current stimulus preceding the spike. A value near zero indicates little similarity between the current that evoked a spike and the STA. Two-dimensional linear-nonlinear model: d The first two eigenvectors (Mode 1 and Mode 2) extracted from the spiketriggered covariance difference matrix (see Methods, Eq. (1). Inset: plot of
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the final covariance difference matrix, blue is lowest value, red is highest value). e Individual spike-evoking stimuli were projected onto each mode, as in c. The projections for all spikes are shown as the joint distribution P(s1,s2|spike) and the marginal distributions P(s1|spike) and P(s2|spike) f Example eigenvalue spectrum corresponding to the covariance difference matrix in d. There is one eigenvalue for each dimension of the covariance difference matrix (500 eigenvalues total, for 50 milliseconds of stimulus history, sampled at 10 kHz.) The two eigenvalues with the greatest magnitude corresponding to Mode 1 (black) and Mode 2 (gray) are shown as open markers. g Example GLM components: stimulus filter k (upper panel) and spiking history filter g
J Comput Neurosci
as estimated through standard methods (Aguera y Arcas et al. 2003; Pillow and Simoncelli 2003; Powers et al. 2005; Pillow et al. 2008). Several methods have been proposed to deal with these regularizing effects. One is to restrict the analysis only to spikes that are well-isolated from previous spikes such that spike history effects are minimized (Aguera y Arcas et al. 2003; Powers et al. 2005). More generally, the spike response model (Kistler et al. 1997) allows the stimulus dependence to vary with the time to the last spike. The autoregressive moving average (ARMA) method predicts the firing rate as a linear weighting of stimulus and spike history (Powers et al. 2005). In the generalized linear model (GLM), the probability of a spike depends nonlinearly on a linear combination of stimulus, self-activity, noise and network factors (Truccolo et al. 2005). When the nonlinearity is chosen from the exponential family, the linear coefficients of such models can be fit efficiently using maximum likelihood methods (Paninski et al. 2004). While the GLM has been successfully applied to capture spike train statistics in several systems (Truccolo et al. 2005; Pillow et al. 2008), it has two potential limitations. The first is that it generally only uses one stimulus filter, and the responses of single neurons are often well modeled in terms of multiple features (Aguera y Arcas et al. 2003; Slee et al. 2005; Hong et al. 2007). The second is that one may wish to avoid restricting the form of the nonlinearity to a specific parametric family, as is allowed by reverse correlation (Aguera y Arcas et al. 2003) or in the method of maximally informative dimensions (Sharpee et al. 2004). In this paper, we compare several approaches to capturing the joint role of stimulus and spike history in predicting spike timing in single neurons. We focus on hypoglossal motoneurons, which have a strong regularizing afterhyperpolarization (AHP). We compare a pure one- or two-dimensional reverse correlation approach to models that incorporate spike history, including the GLM and an alternative in which spike history multiplicatively modulates the instantaneous spike probability (Berry and Meister 1998). This feedback term effectively regularizes the predicted spike trains from a reduced model of quasi-periodically firing motoneurons. In principle, this method permits a broader class of model systems than the GLM. We consider how the different models change with firing rate, driven both by changing the DC input and by changing the stimulus variance. We examine the effect of the AHP current on feature selectivity for different inputs by contrasting the results with those obtained from a neuron model with the AHP eliminated. Finally, we use information theory to quantify how spike timing regularization by added DC and by currents such as the afterhyperpolarization diminished the information transmitted about the fluctuating stimulus.
2 Methods and materials 2.1 Slice preparation The general experimental and surgical procedures used here have been detailed previously (Sawczuk et al. 1995, 1997; Powers et al. 2005; Zeng et al. 2005). Experiments were carried out in accordance with the animal welfare guidelines in place at the University of Washington. Sprague–Dawley rats (2–3 weeks old) of either sex were anesthetized by an intramuscular injection of 1.8 ml/kg of a 5:1.6:6.6 solution of ketamine:xylazine:saline. When fully anesthetized, the animals were decapitated. The skull was exposed at the brainstem level and the underlying brainstem was isolated by removal of the overlying bone, meninges, and cerebellum. A section of brainstem was removed and glued to a Plexiglas tray filled with cooled, modified, artificial cerebrospinal fluid (ACSF). A series of transverse slices 300–400 μm thick were then cut throughout the length of the hypoglossal nucleus, transferred to a holding chamber and incubated at room temperature (19-21 °C for 30 min in the modified ACSF for 30 min, followed by 30 min incubation in standard ACSF. To minimize neural activity during the initial preparation of the slices, two different modified ACSF solutions are used. For slices obtained from younger animals (14–16 days), we use a low Ca2+, high Mg2+ solution (Low-Ca2+ ASCF (in mM): 132 NaCl, 3 KCl, 125 NaH2PO4, 26 NaHCO3, 5 MgCl2, 1 CaCl2 and 10 D-glucose), whereas in older animals (14–42 days) we use a sucrose-based solution (S-ACSF: same as Low-Ca2+ ACSF, except 220 mM sucrose substituted for NaCl, and concentrations of MgCl2 and CaCl2 both 2 mM). Kynurenic acid (1 mM) and sodium lactate (4 mM) are also added to the initial incubation medium to improve cell viability. The standard ASCF was identical to that of the S-ACSF except that 132 mM NaCl was substituted for sucrose. The pH of the ASCF solutions ranged from 7.3 to 7.4 and their osmolalities from 310 to 320 mOsm. 2.2 Measurement and recording techniques Intracellular sharp electrode and whole-cell patch recordings were made from hypoglossal motoneurons. For the whole-cell patch recordings we used a pipette solution containing (mM): 146 KCH3SO4, 5 KCl, 2 MgCl2, 2 EGTA, 10 MOPS, 2 Na2ATP, 0.2 Na3GTP, with KOH / HCl added to bring the pH to 7.2. The osmolality of this solution was typically 310 mOsm. Patch solution aliquots were stored at −20 °C until time of use. The sharp electrode solution contained 0.5 M K-citrate and 1.5 M KCl. Hypoglossal motoneuron identity was based on anatomical location and the similarity of intrinsic properties to those previously reported (Sawczuk et al. 1995, 1997; Powers et al. 2005; Zeng et al. 2005).
J Comput Neurosci
Current-clamp recordings were obtained with either an Axon Instruments Axoclamp 2B amplifier in either bridge or discontinuous current-clamp mode or an Axon Instruments Multiclamp in bridge mode. For whole-cell recordings, published procedures were used to correctly compensate capacity transients and correct for series resistance (Jackson 1992). We selected neurons with resting potentials more negative than −60 mV and action potentials with positive overshoots. If necessary, up to 300 pA of holding current was applied to keep the resting potential below −60 mV, but we only studied cells that exhibited stable repetitive discharge in response to our injected noise stimuli. The cell’s responses to a series of 500 ms depolarizing and hyperpolarizing current pulses of different magnitude were used to determine input resistance, as well as the minimum depolarizing current needed to elicit a spike (i.e., rheobase). After these initial measurements, we repeatedly injected our pre-computed synthetic current waveforms into the cells by a D/A converter with a sampling rate of 10 kHz. These waveforms were composed of a series of brief, hyperpolarizing pulses (used to determine the passive impulse response), followed by a long (38 s) DC step with superimposed filtered Gaussian noise with a standard deviation (STD) of 0.25 nA and a time constant of 1 ms (Powers et al. 2005). Figure 1a shows example noise-driven voltage responses from an in vitro motoneuron recording and the simulated motoneuron. When possible, the DC level of the noise was varied to obtained several different mean firing rates. We collected several trials at a given DC bias, providing up to 30,000 spikes to obtain converged estimates of the covariance matrix. We found that stable estimates of the STA can be obtained from
