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Conceived and designed the experiments: IHS BC MS KPK. Performed the experiments: IHS BC. Analyzed the data: IHS BC. Wrote the paper: IHS BC MS KPK.

The authors have declared that no competing interests exist.

Neurons in the sensory system exhibit changes in excitability that unfold over many time scales. These fluctuations produce noise and could potentially lead to perceptual errors. However, to prevent such errors, postsynaptic neurons and synapses can adapt and counteract changes in the excitability of presynaptic neurons. Here we ask how neurons could optimally adapt to minimize the influence of changing presynaptic neural properties on their outputs. The resulting model, based on Bayesian inference, explains a range of physiological results from experiments which have measured the overall properties and detailed time-course of sensory tuning curve adaptation in the early visual cortex. We show how several experimentally measured short term plasticity phenomena can be understood as near-optimal solutions to this adaptation problem. This framework provides a link between high level computational problems, the properties of cortical neurons, and synaptic physiology.

The excitabilities of individual neurons fluctuate over timescales ranging from milliseconds to hours due to effects such as electrical stimulation

Here we consider adaptation as an estimation problem where neurons attempt to produce stable responses in the presence intrinsic fluctuations. That is, neurons must distinguish changes in sensory stimuli from fluctuations in the excitability of presynaptic neurons. This distinction may be possible if excitabilities and sensory stimuli change over time in different ways. If these two sources of fluctuations in presynaptic activity can be distinguished the noise introduced by excitability fluctuations can be removed. Given observations of presynaptic activity, we first consider the statistical problem of estimating presynaptic excitability. We then assume that, in order to reliably represent sensory drive, the postsynaptic response is the presynaptic activity normalized by the estimated excitability. In many ways this model provides an instantiation of normalization theories of adaptation where the nervous system attempts to correct low-level abnormal responses

Using this excitability estimation framework we examine a range of physiological adaptation phenomena. We examine short-term synaptic depression at a single synapse and medium-term tuning curve adaptation in early visual cortex. Experimental results in both these domains are well-described by a model that implements excitability estimation at the level of single synapses.

Recently, neuronal adaptation has been treated as a mechanism that allows the nervous system to accurately represent stimuli in the face of changes in the statistics of the external world

The central problem in estimating intrinsic fluctuations in excitability is that firing rate information is ambiguous. High firing rates may occur because of strong sensory drive or, alternatively, because presynaptic neurons are highly excitable. In order to adapt in a way that preserves sensory information, the nervous system needs to resolve this ambiguity. Specifically, the nervous system can use information about the way excitability typically changes over time and information about the way sensory drive typically changes over time to estimate presynaptic excitability from presynaptic activity. Here we assume that excitability drifts on multiple timescales around a steady state point

A) Examples of fluctuations in excitability on multiple timescales. We assume that excitability fluctuates due to multiple causes – both slowly fluctuating (e.g. oxygen concentration) and quickly fluctuating (e.g. the activities of neighboring neurons). B) Example of sensory drive. We assume that sensory drive is sparse – non-zero values are relatively rare – and independent from one time-step to the next. C) A schematic depicting the relationship between the excitability, sensory drive, and presynaptic activity. At each time presynaptic activity s is observed. We assume that this activity is the product of the two hidden variables: sensory drive d and excitability (gain) g. Here we assume that the excitability at each time depends on the excitability at the previous time-step. Using this model we can estimate excitability given observations of presynaptic activity, and subsequently normalize postsynaptic responses by this excitability to produce a more stable output.

Given observations of noisy presynaptic activity, our adaptation model estimates the excitability of the presynaptic neuron on each timescale using approximate Bayesian inference (an assumed density filter). We then model the response of the postsynaptic neuron by the observed presynaptic activity divided by the total estimate of the presynaptic excitability (see

First, using a simulation of a single synapse, we illustrate that estimating presynaptic excitability and normalizing postsynaptic responses by these estimates makes neural output more stable (

A simulated neuron receives input from an orientation tuned neuron whose excitability is fluctuating. A) Estimated pre-synaptic fluctuations on three timescales (slow tau = 5 min, intermediate tau = 500 ms, and fast tau = 50 ms). B) The total pre-synaptic gain in this simulation and the optimal estimate given noisy observations of the pre-synaptic activity. C) The post-synaptic response – presynaptic activity normalized by the estimated gain – to a single orientation with and without adaptation. Boxes denote the inter-quartile range; whiskers denote 1.5 times the inter-quartile range. Outliers have been removed for clarity. By cancelling out fluctuations in pre-synaptic excitability the adaptation model can substantially reduce response variability.

At the level of individual synapses, what properties would be required to approximate optimal adaptation? The adaptation rule we present here, normalizing by estimated presynaptic excitability, is based solely on statistical descriptions of how sensory drive and presynaptic excitability change over time. However, one of the main characteristics of this rule is that synaptic strength increases slowly in the absence of presynaptic activity and decreases quickly in the presence of presynaptic activity. These effects roughly correspond to biophysical descriptions of synaptic depletion and recovery. Indeed experimentally observed properties of short-term synaptic depression

A) Low-pass filtering with short-term synaptic depression. Steady-state EPSC size (as a fraction of control) for real data, adapted from

In addition to neural responses at a single synapse, the excitability estimation model may also be used to describe extracellular responses to the repeated stimuli used in typical physiological experiments. One particularly well studied system for this type of experiment is primary visual cortex. Here, recent experiments have found that orientation-selective neurons, when adapted with a stimulus of one orientation, shift their preferred stimulus orientation away from the adapting stimulus

These physiological adaptation effects follow interesting temporal profiles. If the adapting stimulus lies on one flank of the tuning curve, then the response at this “near flank” is quickly reduced. Only later is an increased response observed on the far flank. Interestingly, this far-flank facilitation often resulted in an increase in the magnitude of the response at the (shifted) peak of the tuning curve. The response profile was not merely translated away from the adapting stimulus, but instead underwent changes which occur on at least two separate time scales

Using a feed-forward network of orientation tuned neurons with adapting synapses (

A) The stimuli presented during the control (upper) and adaptation (lower) epochs of the sensory adaptation simulations; these are meant to replicate the stimuli used in Dragoi et al. B) A schematic of the network model used to simulate orientation adaptation. A population of presynaptic neurons, each with its own orientation tuning curve, reacts to the stimulus. These presynaptic activities are then modulated by the presynaptic excitability, weighted, and summed to produce the postsynaptic response. While the synaptic weights remain constant, the excitability estimates are updated over time according to the model. C) The time course of two sets of gains during a period of adaptation to a single stimulus. The gains correspond to presynaptic inputs whose preferred orientations are on the near (blue) and far (yellow) flanks of the control tuning curve. The dashed lines indicate the time points at which the control tuning curve was measured (black), as well as three successive adapted tuning curves (red, gold, and green). D) The control tuning curve (black), as well as three successive adapted tuning curves corresponding to the time points indicated in A.

Interestingly, Dragoi et al.

Control and adapted tuning curves from two real example neurons, adapted from Dragoi et al. (2000) (top). Control and adapted tuning curves from two model neurons whose initial tuning weights were modified to match the electrophysiological data (bottom). In these two cases, the network tuning parameters are slightly different, but the estimation model and parameters are the same.

Finally, it is important to ask how robust the assumptions made under the excitability estimation model are. Since a variety of biological factors (oxygen concentration, neuromodulators, etc) appear to affect pre-synaptic excitability, the assumption of multi-timescale fluctuations seems reasonable. Sampling from the generative model, this approach stably transmits sensory drive with 88±0.3% variance explained. Although the assumed density filter used here assumes that sensory drive is sparse and temporally uncorrelated, we can examine how well it performs when the drive does have temporal structure. Using 1/f noise

Here we have shown that several short term and medium term adaptation effects are consistent with a strategy whereby the nervous system attempts to compute reliably in the presence of constantly changing intrinsic excitabilities. Both short-term adaptation phenomena, those occurring over tens or hundreds of milliseconds

There is a long history of normalization models in psychophysics. These models

More recently, generalizations of normalization-type models have proposed that the nervous system adapts to optimize the amount of information transmitted by a sensory system which is limited by noise or the availability of neural resources

However, several recent papers have considered information maximization at the cellular or synaptic level

A number of computational roles for synaptic depression have been suggested such as decorrelating inputs

Synaptic depression has also been considered as a mechanism for a number of cellular-level phenomena including direction selectivity and contrast adaptation

In explaining medium timescale adaptation phenomena we have focused on “repulsive” adaptation, where tuning curves are shifted away from an adapting stimulus. At the perceptual level, examples of repulsive adaptation include the tilt after-effect

In the model presented here, the assumption of sparse, temporally uncorrelated sensory drive is unlikely to reflect the true statistics of an external variable. Natural stimuli are spatially and temporally correlated on a range of scales

More generally, the model described here uses a simple linear response model and thus clearly only implements a rough approximation to the problem solved by the nervous system. For example, the inputs in the model are independent of one another, and the excitabilities of each presynaptic neuron are estimated and adapted separately. This framing ignores the correlations which are known to exist in the firing patterns of neural populations, and such correlations may be crucial in explaining other aspects of physiological adaptation phenomena. In addition, the simple network model that we use to simulate orientation tuning omits many of the properties which such thalamo-cortical networks are known to exhibit. We chose the simple model here as it allows us to compactly solve the statistical problem of estimating excitability. However, stable computation is an appealing general principle.

In a statistically optimal system adaptation is determined by assumptions about the way both the nervous system and sensory drive change over time. When these assumptions are violated, for example by experiments that repeatedly present stimuli that are rare in the natural environment, phenomena such as the tilt and motion after-effects are the result. Underlying our analysis of adaptation is the assumption that the stimuli which lead to effects such as the tilt aftereffect and the motion aftereffect are in fact very rare in the natural environment – and the price the nervous system pays for adaptation under normal situations is thus very small. These unusual stimuli fool the perceptual system by mimicking a situation in which the excitability of some neurons has been increased (and others possible decreased). We propose that physiological and perceptual sensory adaptation stems from this fundamental ambiguity that exists between the intensity of sensory stimuli and the excitability of the neurons that process these signals.

We assume that a post-synaptic neuron seeks to estimate fluctuations in its inputs that occur over time scales ranging from a few milliseconds to minutes. To accomplish this goal, the neuron must estimate the excitability fluctuations of its inputs at each of these time scales. We refer to these estimates of excitability as gains.

Throughout the following, we use

We have described a statistical model whose conditional independence structure is equivalent to that of the state space model, a standard framework for describing the dynamics of normally distributed variables. However, the likelihood distribution for

Given the assumptions about how

To illustrate the role of adaptation in stable computation, we simulate from the generative model described above where both fluctuations in presynaptic excitability and the sensory drive are known (

We then use this model of gain dynamics to model the short-term changes in strength of a single synapse. We compare the resulting gain changes to a model based on neurotransmitter depletion

To fit the data from

To model results for

We also apply the model of gain dynamics to a simple network model of orientation tuning in primary visual cortex. In this model, a single postsynaptic cell receives input from _{i}

At each time step, the stimulus causes each of the presynaptic cells to become active according to its tuning curve. This stimulus drive is in addition to a lower, spontaneous level of activity

We should note that, in addition to the pre-adaptation tuning curve, the tuning widths of the presynaptic neurons affect the specific time-course and shape of adapted tuning curves. The input tuning widths for the two simulations in

Each orientation adaptation simulation is divided into two epochs (

Again, as with the simulations of single synapses we assume that the true presynaptic excitability is fixed. In this case the estimated presynaptic excitability of each synapse is very different from the true presynaptic excitability. Instead, presynaptic activity which is unlikely to be generated by sensory drive, given our assumptions, is attributed to fluctuations in presynaptic excitability.

The assumed density approximation in 2D. Calculating the true posterior at each time-step is difficult to do analytically, since the likelihood is non-Gaussian (left column). However, we can approximate the posterior with a Gaussian at each time-step (right column, ADF). Here we show a single time-step of this approximation for 2-dimensions, where s = 2, cov_t = 2.5I, and g(t-1) = [0.5; 0.5] (top row) or g(t-1) = [1.5; 1.5] (bottom row). The blue line denotes the maxima of the likelihood, the blue circle denotes the maximum of the prior, and the black cross denotes the maximum of the posterior.

(0.74 MB EPS)

Gain control. The excitability estimation model naturally reproduces many aspects of gain control. Input rates are naturally normalized by ongoing activity. In this simulation a neuron receives two synaptic inputs with average rates of 100Hz and 10Hz. (left) 50% modulation of the 100Hz input, produces ∼12% modulation in the output with adaptation and ∼50% modulation without adaptation. (middle) 50% modulation of the 10Hz input, produces ∼15% modulation in the output with adaptation and 5% without modulation. (right) 5% modulation of the 100Hz input produces <5% modulation in the output with adaptation. Blue and green denote the rates of the 100Hz and 10Hz input respectively.

(3.12 MB EPS)

Assumed density filtering.

(0.04 MB DOC)

Excitability estimation and gain control.

(0.03 MB DOC)

Thanks to Ning Qian, Máté Lengyel, and Peter Dayan for helpful comments.