A model for non-monotonic intensity coding

Peripheral neurons of most sensory systems increase their response with increasing stimulus intensity. Behavioural responses, however, can be specific to some intermediate intensity level whose particular value might be innate or associatively learned. Learning such a preference requires an adjustable trans- formation from a monotonic stimulus representation at the sensory periphery to a non-monotonic representation for the motor command. How do neural systems accomplish this task? We tackle this general question focusing on odour-intensity learning in the fruit fly, whose first- and second-order olfactory neurons show monotonic stimulus–response curves. Nevertheless, flies form associative memories specific to particular trained odour intensities. Thus, downstream of the first two olfactory processing layers, odour intensity must be re-coded to enable intensity-specific associative learning. We present a minimal, feed-forward, three-layer circuit, which implements the required transformation by combining excitation, inhibition, and, as a decisive third element, homeostatic plasticity. Key features of this circuit motif are consistent with the known architecture and physiology of the fly olfactory system, whereas alternative mechanisms are either not composed of simple, scalable building blocks or not compatible with physiological observations. The simplicity of the circuit and the robustness of its function under parameter changes make this computational motif an attractive candidate for tuneable non-monotonic intensity coding.

2. Introduction

Varying a sensory stimulus can influence behaviour in two fundamentally different ways. First, the map from stimulus to behaviour can be one-to-one. For example, the reaction time of human beings to a light stimulus decreases steadily with increasing light intensity [1]. At the neuronal level, monotonic stimulus–response curves suffice to explain this phenomenon. Second, a particular behaviour may only be triggered by a certain range of intermediate stimulus values; for instance, rats and fruit flies prefer weak, but not strong, salt solutions over plain water [2,3]. In this case, the brain needs to represent the stimulus in a non-monotonic way to generate the appropriate behaviour.

For some stimulus attributes, bell-shaped tuning curves at the sensory periphery solve this task. The peaked frequency tuning of hair cells [4], for example arises because the basal membrane of the vertebrate cochlea vibrates most strongly at a location determined by the frequency of the presented sound. For other stimulus dimensions, such as sound amplitude [5], sensory neurons have monotonic input–output curves, raising the question of how non-monotonic stimulus dependencies of behavioural responses are generated.

A suitable system to study this general question is odour-intensity learning. Odour intensity is typically encoded in a monotonic way by the first- and second-order olfactory neurons; consequently, the neuronal population activated by an odour grows with increasing odour intensity and the representation of a lower intensity is nested within that of a higher intensity (e.g. [6–8]). The overall increase in neuronal activation with rising odour intensity can be useful to explain the ability to detect odour gradients (as argued, e.g. in [9–11]) as well as the improvement of olfactory detection, associative learning and memory retrieval at higher intensities (e.g. [12]). Changes in the hedonic value of an odour with increasing intensity can arise if neurons with different sensitivities are connected to opponent downstream pathways (e.g. [13]). Finally, changes in discriminability across odours with rising intensity are consistent with growing odour representations (e.g. [14]). However, a key behavioural observation remains unexplained: animals form associative memories specific to trained odour intensities such that later on, neither lower nor higher intensities release as strong a conditioned behaviour, as shown in the fruit fly [12,15–17], honeybee [18] and mouse [19]. This intensity specificity of learning suggests that along the olfactory pathway, downstream of the initial monotonic encoding, odour intensity must be re-coded in a non-monotonic manner.

We present a simple, biologically plausible neuronal circuit motif that does just this. We quantitatively compare the intensity coding ability of this model to the intensity specificity of olfactory memories, as assayed in the fruit fly and discuss how this circuit may be implemented in the fly olfactory system, thus leading to experimentally testable hypotheses. The circuit motif found may also be relevant for other cases where stimulus intensity must be encoded in a non-monotonic fashion to enable intensity-specific behaviours (for an example in the auditory modality, see [5]).

3. Material and methods

3.1 Input layer

The activity of excitatory and inhibitory input neurons (figure 2b) are described by logistic input–output functions:

3.1

where i is the odour intensity in logarithmic units. Thus, a_k and a_inh are the odour intensities at the turning points of the respective logistic sensitivity functions, i.e. a large negative a-value implies a high sensitivity. The factor 4 in the exponents is chosen so that b and b_inh are the slopes at the turning points, where b>b_inh. The parameter inh_max>1 scales the sensitivity function of the inhibitory input neuron. For simplicity, only three excitatory input neurons are considered. Their a_k values are shifted in steps of one logarithmic unit.

3.2 Intermediate layer

The activity of the intermediate-layer neurons (figure 2c) are calculated as rectified weighted sums of the input activities as

3.2

where w_exc and w_inh are the weights of the respective excitatory and inhibitory inputs. The rectifying function Rect(x) is defined as Rect(x<0)=0 and Rect(x≥0)=x, and results in a threshold neuronal activation function.

3.3 Homeostatic plasticity

We consider two scenarios for homeostatic plasticity. In both cases, we do not model how the synaptic strength changes in response to each individual stimulus presentation but rather calculate the resulting mean effect of homeostatic plasticity under the assumption that already prior to the specific associative odour-shock training, the system has been exposed to odours drawn from a broad range of concentrations.

In the first scenario (figure 3a(i)), the weights of the inhibitory synapses to the intermediate layer are set uniformly to be w_inh=−1; whereas each excitatory synapse (w_exc) is subject to homeostatic plasticity. To implement the mean effect of this regulatory process, the weights w_exc are adjusted based on the sensitivity of the respective excitatory input neurons: the more sensitive an input neuron is (more negative a-value), the higher its mean activation and, consequently, the mean rate at which it drives the downstream intermediate-layer neuron. This effect will be balanced by homeostatic plasticity. As a measure of the input neuron's sensitivity, we take the integral of the input–output function exc_k(i) over a concentration range [c₀, c₁]:

3.3

The sensitivity function s(a) approaches c₁–c₀ for and zero for . For intermediate values c₀<a<c₁, s(a) scales roughly linear in a. Then, based on s(a), we adjust the respective excitatory output weight as

3.4

where α is a scaling factor and d is set such that w_exc(a)>0 (see inset in figure 3a(i)). Thus, in the spirit of homeostatic plasticity, the more sensitive an excitatory input neuron is, the weaker its synapse to the intermediate layer will be. Mechanistically, this could either be implemented through ‘local’ homeostatic plasticity [20,21] acting directly at this excitatory synapse, or through classical homeostatic plasticity [22], as we only consider a single excitatory input to each intermediate-layer neuron.

In the second scenario (figure 3b(i)), the weights of all excitatory synapses are set to w_exc=1. Implementing the mean effect of homeostatic plasticity, the weight of each inhibitory synapse is scaled according to the sensitivity of the cognate excitatory input neuron as

3.5

where is a scaling factor (see inset in figure 3b(i)). Thus, the smaller the excitatory drive of an intermediate neuron, the weaker is also its inhibitory input, in accordance with experimental findings on homeostatic plasticity at inhibitory synapses [21,23,24].

3.4 Output neuron and associative plasticity

The activity of the output neuron is calculated as the weighted sum of the intermediate-layer neuron activities:

3.6

Initially, the weights w_training,k are all zero. During associative odour-shock training (e.g. figures 2d, 3a(iii),b(iii)), these weights change proportional to the odour-induced activity in the respective intermediate neuron, owing to the delivery of a concurrent reinforcement signal as

3.7

where θ(x) is the Heaviside function, defined as 0 if x≤0 and 1 otherwise, representing the presence versus absence of shock and i_training is the odour intensity at training.

4. Results

4.1 Olfactory memories of flies are odour-intensity specific

The intensity specificity of fruit fly olfactory memories has been reported in several studies using different developmental stages, experimental rationale and reinforcers ([12,15–17,25,26]; for a comparative discussion see [12]). We start with a meta-analysis of three experiments that apply a common paradigm to three odours [12]. In each experiment, flies are trained en masse, with pairings of a chosen intensity of the respective odour and electric shock. Different groups of flies are then tested for their avoidance of this odour at the trained, a lower or a higher intensity (figure 1a). In each case, conditioned avoidance is scored by comparing the behaviour of flies trained as explained with paired presentation of odour and shock versus flies trained with temporally unpaired presentation of the same stimuli; thus, the scores refer to effects of associative learning and not to innate odour-responsiveness. Across all three experiments, flies show the strongest conditioned avoidance when the testing and training intensities match (figure 1b). For better comparison across experiments, we align the three datasets along the stimulus and the response axes and find similar Gaussian fits, despite the diversity of odours (figure 1c). Results from an appetitive olfactory learning assay in Drosophila larvae [17] paint a similar picture (see the legend of figure 1c).

4.2 A simple circuit motif for odour-intensity-specific memories

Fruit fly olfactory sensory neurons (OSNs) and projection neurons (PNs) increase their activity with rising odour intensity at the single-cell level, as exemplified in figure 1e,f (see also [8,10,27,29–32] for demonstration of this property using a variety of methods). As a direct consequence of such monotonic input–output curves, an odour at low intensity excites relatively few neurons, whereas the same odour at a higher intensity recruits not only these but also additional neurons. Based on such a nested representation of odour intensities alone, the memory trace of a low-intensity odour would be activated at least as strongly by a higher intensity of the same odour. However, olfactory associative memories in flies are intensity specific (figure 1a–c). This implies that non-monotonic intensity responses must emerge in downstream layers of the olfactory pathway. The following model accomplishes this task.

The input layer of the model harbours multiple excitatory neurons (figure 2a, blue) with different, monotonic responses, represented by logistic functions that are shifted by different offsets along the stimulus axis (figure 2b, blue). These functions are inspired by fly OSN- and PN-electrophysiology (e.g. [27] and [29]; figure 1e,f), as well as computational models of olfactory transduction (e.g. [33]). The modelled excitatory input neurons are connected one-to-one with neurons of the intermediate layer (figure 2a, green). In addition, a single inhibitory neuron with monotonic input–output function (figure 2a,b, red) provides input to all intermediate neurons. The convergence of excitation and inhibition endows each intermediate neuron with a bell-shaped tuning curve (figure 2c). The relative shift of sensitivity across the excitatory input neurons (figure 2b, blue) and the shallower sensitivity curve of the inhibitory neuron as compared with the excitatory neurons cause the intermediate neurons to differ in their tuning curves (figure 2c) but the nestedness of these tuning curves rules out that memories are intensity specific.

To illustrate this important limitation we introduce an output neuron, onto which all intermediate neurons converge (figure 2a, black). Prior to any training, the synaptic weights are set to zero so that the output neuron does not respond to even the most intense odour (note that the innate olfactory behaviour pathway is not represented in the model). When we train the circuit by pairing a given odour intensity with electric shock (figure 2d, training), each intermediate neuron is activated to a certain degree, which depends on its tuning curve and the intensity of the presented odour. In addition, a reinforcement signal, induced by the electric shock is delivered to the output synapse of each intermediate neuron (figure 2a, yellow). Owing to this reinforcement signal, each output synapse is strengthened proportional to the respective level of odour-induced activity (figure 2d, training). This potentiation of output synapses is the trace for the odour-shock memory. To read out this trace at test, we present the circuit with various odour intensities and measure the activity of the output neuron (figure 2d, test). If the circuit mimics the flies' intensity-specific learning (figure 1a–c), the output neuron will respond most strongly when the training and testing intensities match. This is not the case (figure 2d, test): although after all three kinds of training, the output neuron activity depends on the odour intensity with a bell-shaped function, the peaks do not correspond to the respective odour intensities used at training.