. Many types of neurons exhibit subthreshold resonance. However, little is known about whether this frequency preference influences spike emission. Here, the link between subthreshold resonance and firing rate is examined in the framework of conductance-based models. A classification of the subthreshold properties of a general class of neurons is first provided. In particular, a class of neurons is identified in which the input impedance exhibits a suppression at a nonzero low frequency as well as a peak at higher frequency. The analysis is then extended to the effect of subthreshold resonance on the dynamics of the firing rate. The considered input current comprises a background noise term, mimicking the massive synaptic bombardment in vivo. Of interest is the modulatory effect an additional weak oscillating current has on the instantaneous firing rate. When the noise is weak and firing regular, the frequency most preferentially modulated is the firing rate itself. Conversely, when the noise is strong and firing irregular, the modulation is strongest at the subthreshold resonance frequency. These results are demonstrated for two specific conductance-based models and for a generalization of the integrate-and-fire model that captures subthreshold resonance. They suggest that resonant neurons are able to communicate their frequency preference to postsynaptic targets when the level of noise is comparable to that prevailing in vivo.

Badel L, Lefort S, Brette R, Petersen CC, Gerstner W, Richardson MJ. Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. J Neurophysiol 99: 656 -666, 2008. First published December 5, 2007 doi:10.1152/jn.01107.2007. Neuronal response properties are typically probed by intracellular measurements of current-voltage (I-V) relationships during application of current or voltage steps. Here we demonstrate the measurement of a novel I-V curve measured while the neuron exhibits a fluctuating voltage and emits spikes. This dynamic I-V curve requires only a few tens of seconds of experimental time and so lends itself readily to the rapid classification of cell type, quantification of heterogeneities in cell populations, and generation of reduced analytical models. We apply this technique to layer-5 pyramidal cells and show that their dynamic I-V curve comprises linear and exponential components, providing experimental evidence for a recently proposed theoretical model. The approach also allows us to determine the change of neuronal response properties after a spike, millisecond by millisecond, so that postspike refractoriness of pyramidal cells can be quantified. Observations of I-V curves during and in absence of refractoriness are cast into a model that is used to predict both the subthreshold response and spiking activity of the neuron to novel stimuli. The predictions of the resulting model are in excellent agreement with experimental data and close to the intrinsic neuronal reproducibility to repeated stimuli. I N T R O D U C T I O NAccurate models of electrically active cells and their interactions are central requirements for the understanding of the computational processes taking place in nervous tissue. The construction of network models, even at the level of cortical columns, requires the identification of cell classes and the quantification of both their typical behavior and the heterogeneities within a population. The volume of data that is required for this tissue-level modeling demands a high-throughput approach in which response properties can be routinely measured.Electrophysiology provides an array of techniques for the extraction of neuronal response properties. Standard methods involve probing the response to step-change stimuli leading to current-voltage (I-V) curves for the steady-state or instantaneous response. Used systematically with pharmacology, they can yield a full conductance-based description Koch 1999) although the time required is prohibitive for routine neuron-by-neuron classification. More recently, an elegant optimization method (Huys et al. 2006) has been proposed that promises to significantly facilitate the construction of biophysically detailed models, given some prior knowledge of the kinetics of the channels present.Detailed models, comprising hundreds of compartments, are important for understanding the biophysical properties probed during electrophysiological and pharmacological manipulations. Such models can be used for network simulatio...

A neuron in an active cortical circuit is subject to a fluctuating synaptic drive mediated by conductance changes. It was recently demonstrated that synaptic conductance effects in vivo significantly alter the integrative properties of neurons. These effects are missed in models that approximate the synaptic drive as a fluctuating current. Here the membrane-potential distribution and firing rate are derived for the integrate-and-fire neuron with delta correlated conductance-based synaptic input using the Fokker-Planck formalism. A number of different input scenarios are examined, including balanced drive and fluctuation changes at constant conductance, the latter of which corresponds to shifts in synchrony in the presynaptic population. This minimal model captures many experimentally observed conductance-related effects such as reduced membrane-potential fluctuations in response to increasing synaptic noise. The solvability of the model allows for a direct comparison with current-based approaches, providing a basis for assessing the validity of existing approximation schemes that have dealt with conductance change. In particular, a commonly used heuristic approach, whereby the passive membrane time constant is replaced by a drive-dependent effective time constant, is examined. It is demonstrated that this approximation is valid in the same limit that the underlying diffusion approximation holds, both for delta correlated as well as filtered synaptic drive.

Integrate-and-fire models are mainstays of the study of single-neuron response properties and emergent states of recurrent networks of spiking neurons. They also provide an analytical base for perturbative approaches that treat important biological details, such as synaptic filtering, synaptic conductance increase, and voltage-activated currents. Steady-state firing rates of both linear and nonlinear integrate-and-fire models, receiving fluctuating synaptic drive, can be calculated from the time-independent Fokker-Planck equation. The dynamic firing-rate response is less easy to extract, even at the first-order level of a weak modulation of the model parameters, but is an important determinant of neuronal response and network stability. For the linear integrate-and-fire model the response to modulations of current-based synaptic drive can be written in terms of hypergeometric functions. For the nonlinear exponential and quadratic models no such analytical forms for the response are available. Here it is demonstrated that a rather simple numerical method can be used to obtain the steady-state and dynamic response for both linear and nonlinear models to parameter modulation in the presence of current-based or conductance-based synaptic fluctuations. To complement the full numerical solution, generalized analytical forms for the high-frequency response are provided. A special case is also identified--time-constant modulation--for which the response to an arbitrarily strong modulation can be calculated exactly.

The subthreshold membrane voltage of a neuron in active cortical tissue is a fluctuating quantity with a distribution that reflects the firing statistics of the presynaptic population. It was recently found that conductancebased synaptic drive can lead to distributions with a significant skew. Here it is demonstrated that the underlying shot noise caused by Poissonian spike arrival also skews the membrane distribution, but in the opposite sense. Using a perturbative method, we analyze the effects of shot noise on the distribution of synaptic conductances and calculate the consequent voltage distribution. To first order in the perturbation theory, the voltage distribution is a gaussian modulated by a prefactor that captures the skew. The gaussian component is identical to distributions derived using current-based models with an effective membrane time constant. The well-known effective-time-constant approximation can therefore be identified as the leading-order solution to the full conductance-based model. The higher-order modulatory prefactor containing the skew comprises terms due to both shot noise and conductance fluctuations. The diffusion approximation misses these shot-noise effects implying that analytical approaches such as the Fokker-Planck equation or simulation with filtered white noise cannot be used to improve on the gaussian approximation. It is further demonstrated that quantities used for fitting theory to experiment, such as the voltage mean and variance, are robust against these non-Gaussian effects. The effective-time-constant approximation is therefore relevant to experiment and provides a simple analytic base on which other pertinent biological details may be added.

The movements of the human arm have been extensively studied for a variety of goal-directed experimental tasks. Analyses of the trajectory and velocity of the arm have led to many hypotheses for the planning strategies that the CNS might use. One family of control hypotheses, including minimum jerk, snap and their generalizations to higher orders, comprises those that favor smooth movements through the optimization of an integral cost function. The predictions of each order of this family are examined for two standard experimental tasks: point-to-point movements and the periodic tracing of figural forms, and compared both with experiment and the two-thirds power law. The aim of the analyses is to generalize previous numerical observations as well as to examine movement segmentation. It is first shown that contrary to recent statements in the literature, the only members of this family of control theories that match reaching movement experiments well are minimum jerk and snap. Then, for the case of periodic drawing, both the ellipse and cloverleaf are examined and the experimentally observed power law is derived from a first-principles approach. The results for the ellipse are particularly general, representing a unification of the two-thirds power law and smoothness hypotheses for ellipses of all reasonable eccentricities. For complex shapes it is shown that velocity profiles derived from the cost-function approach exhibit the same experimental features that were interpreted as segmented control by the CNS. Because the cost function contains no explicit segmented control, this result casts doubt on such an interpretation of the experimental data.

The synaptic coupling between neurons in neocortical networks is sufficiently strong so that relatively few synchronous synaptic pulses are required to bring a neuron from rest to the spiking threshold. However, such finite-amplitude effects of fluctuating synaptic drive are missed in the standard diffusion approximation. Here exact solutions for the firing-rate response to modulated presynaptic rates are derived for a neuron receiving additive excitatory and inhibitory synaptic shot noise with exponential amplitude distributions. The shot-noise description of the neuronal response to synaptic dynamics is shown to be richer and qualitatively distinct from that predicted by the diffusion approximation. It is also demonstrated how the framework developed here can be generalized to multiplicative shot noise so as to better capture effects of the inhibitory reversal potential.

Inter-pyramidal synaptic connections are characterized by a wide range of EPSP amplitudes. Although repeatedly observed at different brain regions and across layers, little is known about the synaptic characteristics that contribute to this wide range. In particular, the range could potentially be accounted for by differences in all three parameters of the quantal model of synaptic transmission, i.e. the number of release sites, release probability and quantal size. Here, we present a rigorous statistical analysis of the transmission properties of excitatory synaptic connections between layer-5 pyramidal neurons of the somato-sensory cortex. Our central finding is that the EPSP amplitude is strongly correlated with the number of estimated release sites, but not with the release probability or quantal size. In addition, we found that the number of release sites can be more than an order of magnitude higher than the typical number of synaptic contacts for this type of connection. Our findings indicate that transmission at stronger synaptic connections is mediated by multiquantal release from their synaptic contacts. We propose that modulating the number of release sites could be an important mechanism in regulating neocortical synaptic transmission.