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Dendrites of many neuron types exhibit complex morphologies and spatially modulated ionic channel compositions that generate non-trivial integrative properties (Major et al., 2013). These have been described as cascades of linear filters and static nonlinearities (Häusser and Mel, 2003; Larkum et al., 2009). Iso-response methods (see Gollisch and Herz, 2012) provide an alternative approach to understanding the intricate, nonlinear computations in dendrites, while making fewer prior assumptions. Within an iso-response approach, inputs to a dynamical system are varied such that a chosen output measure stays constant. These stimuli define lower-dimensional “iso-response manifolds” whose shape directly reflects the stimulus interaction, regardless of any further nonlinearity that acts on the combined signal. The output measure can range from the neuron’s firing rate (Gollisch et al., 2002) or the probability of a single spike (Gollisch and Herz, 2005) to first-spike latency or even a spike’s phase with respect to some ongoing brain rhythm. As an example (Fig. 1), take a 2D stimulus space parameterized by x1 and x2; the curved surfaces represent the response r(x1,x2) for two alternative models, which take the quadratic (a) and the linear sum (b) as the argument of a sigmoidal nonlinearity. The two scenarios are fundamentally different but generate identical one-dimensional responses. In addition, measurements along radial directions produce similar response curves, as shown by the black lines running along the surfaces. Iso-response curves (r=const), drawn as projections below the surface plots, however, reveal the different underlying processes. Using detailed multi-compartment models, we here show three examples of how the iso-response framework aids in understanding dendritic computations: First, in a pyramidal cell model (Poirazi et al., 2003), we find (Fig. 2) that, in terms of the membrane potential at the soma, activation of synapses on two selected dendritic compartments can give rise to linear (b) as well as curved iso-response lines (a,c). Notably, even inputs to spatially separated compartments (a) can be integrated more non-linearly than nearby inputs (b). Long-range nonlinear interactions are facilitated by local dendritic spikes, which are easily elicited in distal compartments. Second, we find that while the dendritic locations strongly influence the signature of nonlinear interactions, it is less sensitive to changes in the level of background synaptic activity. Finally, we stimulate a Purkinje cell model (de Shutter and Bower, 1994) with a constant input, resulting in regular spiking. We define the output as change in spike timing caused by activating two synapses at different dendritic locations and phases of the regular oscillation. For a large number of parameters we find that the dendrite performs a temporal max-operation on the inputs). These results demonstrate that iso-response methods are well suited to unravel complex dendritic computations in model neurons and that they can provide a powerful tool for experimental research with applications in multi-patch or optogenetic recordings.

Does the nervous system "tune" itself to perform at peak efficiency? Optimal transmission of information in a single nerve cell occurs when the response is matched to the statistics of naturally occurring stimuli, such that all firing rates are used with equal probability and that redundant temporal correlations in the input are removed. Non-Hebbian, local learning rules are developed to adapt the voltage-dependent ionic conductances in Hodgkin-Huxley models of neurons with the goal of matching the neuron's response to the statistics of natural stimuli. These learning rules allow a nerve cell to maximize the amount of information transmitted about arriving stimuli. At a more detailed level, information transmission in neurons is limited by the noise in the input, defined as the root mean square of the fluctuations in the input. Three different performance measures are shown to scale identically as a function of the noise in simple models of neurons that have both a voltage and current threshold. These performance measures are: the probability of correctly detecting a constant input in a limited time, the signal-to-noise ratio in response to sinusoidal input, and the mutual information between an arbitrarily varying input and the output spike train of the model neuron. Of these, detecting a constant signal is the simplest and most fundamental quantity. For subthreshold signals, the model exhibits stochastic resonance, a non-zero noise amplitude that optimally enhances signal detection. In this case, noise paradoxically does not limit, but instead improves performance. Even though the noise amplitude can dwarf the signal, detection of a weak constant signal using stochastic resonance is still possible when the signal elicits on average only one additional spike. Stochastic resonance could thus play a role in neurobiological sensory systems, where speed is of the utmost importance and averaging over many individual spikes is not possible.

We investigate a model for neural activity that generates long range temporal correlations, 1/ f noise, and oscillations in global activity. The model consists of a two-dimensional sheet of leaky integrate-and- fire neurons with feedback connectivity consisting of local excitation and surround inhibition. Each neuron is independently driven by homogeneous external noise. Spontaneous symmetry breaking occurs, resulting in the formation of "hotspots" of activity in the network. These localized patterns of excitation appear as clusters that coalesce, disintegrate, or fluctuate in size while simultaneously moving in a random walk constrained by the interaction with other clusters. The emergent cross-correlation functions have a dual structure, with a sharp peak around zero on top of a much broader hill. The power spectrum associated with single units shows a 1/ f decay for small frequencies and is flat at higher frequencies, while the power spectrum of the spiking activity averaged over many cells-equivalent to the local field potential-shows no 1/ f decay but a prominent peak around 40 Hz.