Computing with neural synchrony

Abstract

Neurons communicate primarily with spikes, but most theories of neural computation are based on firing rates. Yet, many experimental observations suggest that the temporal coordination of spikes plays a role in sensory processing. Among potential spike-based codes, synchrony appears as a good candidate because neural firing and plasticity are sensitive to fine input correlations. However, it is unclear what role synchrony may play in neural computation, and what functional advantage it may provide. With a theoretical approach, I show that the computational interest of neural synchrony appears when neurons have heterogeneous properties. In this context, the relationship between stimuli and neural synchrony is captured by the concept of synchrony receptive field, the set of stimuli which induce synchronous responses in a group of neurons. In a heterogeneous neural population, it appears that synchrony patterns represent structure or sensory invariants in stimuli, which can then be detected by postsynaptic neurons. The required neural circuitry can spontaneously emerge with spike-timing-dependent plasticity. Using examples in different sensory modalities, I show that this allows simple neural circuits to extract relevant information from realistic sensory stimuli, for example to identify a fluctuating odor in the presence of distractors. This theory of synchrony-based computation shows that relative spike timing may indeed have computational relevance, and suggests new types of neural network models for sensory processing with appealing computational properties.

Figure 1. Synchrony receptive field.

A, When neuron A is hyperpolarized by an inhibitory input (top), its low-voltage-activated K channels slowly close (bottom), which makes the neuron fire when inhibition is released (neuron models are used in this and other figures). B, Spike latency is negatively correlated with the duration of inhibition (black line). Neuron B has similar properties but different values for the threshold and K channel parameters (blue line). The synchrony receptive field of neurons A and B is the stimulus with duration 500 ms. C, A postsynaptic neuron receives inputs from A and B. D, It is more likely to fire when the stimulus in the synchrony receptive field of A and B. E, Distribution p(v) of the postsynaptic membrane potential when the neuron is not stimulated (left, “background”) and when it receives an input of size Δv (right, “signal”; e.g. neurons A and B shown in panel C fire together). The standard deviation of the distribution is σ. The neuron fires when v is greater than the spike threshold θ. F, Receiver-operation characteristic (ROC) for three levels of noise, obtained by varying the threshold θ (black curves). The hit rate is the probability that the neuron fires within one integration time constant τ when depolarized by Δv, and the false alarm rate is the firing probability without depolarization. The corresponding theoretical curves, with sensitivity index d′ = Δv/σ, are shown in red. G, When a neuron receives two synchronous inputs of size w (PSP peak), the peak potential is 2w plus the background noise (left). When the second input arrives after a delay δ, the peak is plus the background noise (right). H, Distinguishing between synchronous inputs and delayed inputs corresponds to setting a threshold θ between two distributions separated by .

Figure 2. Decoding synchrony patterns in a heterogeneous population.

Each column corresponds to one stimulus duration. A, Color represents the latency of the spike produced by each neuron responding to the stimulus (white if the neuron did not spike). Thus, neurons with the same color are synchronous for that specific stimulus (duration). The population can be divided in groups of synchronous neurons (i.e., with the same color), forming the “synchrony partition”. Circled neurons belong to the synchronous group of neuron A. B, Each synchronous group projects to a postsynaptic neuron. Each duration is associated with an assembly of postsynaptic neurons. C, Activation of the postsynaptic assembly as a function of duration (grey: individual neurons; black: average).

Figure 3. Generality of coincidence detection.

A, Activation of postsynaptic assemblies as a function of duration (as in Fig. 2C) for three noise levels: σ_v = 0.07, 0.14, 0.28 (bottom to top curve). B, Same as A with synaptic conductances and σ_v = 0.14 (as in Fig. 2C; grey: individual neurons; black: average). C, Same as B using neurons with rebound spiking (identical to the presynaptic neurons).

Figure 4. Learning in the duration model.

A, In addition to homeostasis, synaptic weights are modified by for every pair of pre and postsynaptic spikes at times t_pre and t_post, respectively. B, Presynaptic neurons project to random postsynaptic neurons, with on average 5 synapses per postsynaptic neuron. C, Duration selectivity curves for 5 postsynaptic neurons at the beginning (top) and end (bottom) of the learning period. D, Temporal evolution of the synaptic weights of the neuron corresponding to the blue curves in C. E, Spike latency as a function of stimulus duration for all the presynaptic neurons of the postsynaptic neuron selected in D. Red curves correspond to the two strongest synapses. F, For three postsynaptic neurons (colors as in C), synaptic weights are shown against spike latency of the corresponding presynaptic neurons, at the best duration of the postsynaptic neuron.

Figure 5. Synchrony mechanism with sensory stimuli.

A, Schematic representation of stimulus encoding by a neuron: the stimulus S is filtered through the receptive field N, and the resulting signal N(S) is nonlinearly transformed into spike trains. The synchrony receptive field of two different neurons A and B is the set of stimuli such that the two filtered signals match: N_A(S) = N_B(S). B, Schematic representation of a standard receptive field (N(S)>θ) and a synchrony receptive field in a two-dimensional world. C, Fluctuating input and independent noise. Right: input autocorrelation (time constant 5 ms). D, Responses of a noisy integrate-and-fire model in repeated trials. Right: shuffled auto-correlogram (SAC) for different signal-to-noise ratios (SNR). E, Precision and reliability of spike timing as a function of SNR.

Figure 6. Synchrony without trial-to-trial reproducibility.

A, Neurons A and B receive the same stimulus-driven input, neuron C receives a different one. The stimuli are identical in all trials but all neurons receive a shared input that varies between trials. Each neuron also has a private source of noise. B, Responses of neurons A (black), B (red) and C (blue) in 25 trials, with a signal-to-noise ratio (SNR) of 10 dB (shared vs. private). C, The shuffled autocorrelogram of neuron A indicates that spike trains are not reproducible at a fine timescale. D, Nevertheless, the average cross-correlogram between A and B shows synchrony at a millisecond timescale, which does not appear between A and C. E, Same as D with SNR = 5 dB (note the different vertical scale). F, Same as D with SNR = 15 dB.

Figure 7. Structure and synchrony.

A, Binaural hearing (simplified). The sound arrives at the two ears after a propagation delay d_L and d_R. Monaural neurons A and B project to a binaural neuron with axonal conduction delays δ_L and δ_R. Synchrony (seen on the postsynaptic side) occurs when d_R−d_L = δ_L−δ_R, corresponding to a specific interaural time difference. B, Pitch. Two monaural neurons responding to a sound project to a postsynaptic neuron with axonal delays δ_A and δ_B. From the postsynaptic point of view, synchrony occurs for a periodic sound with period 1/f₀ matching the delay difference: 1/f₀ = δ_B−δ_A. C, Olfaction. Left, Odor concentration fluctuates rapidly because of turbulences, and odorant molecules bind to different types of receptors. Each receptor has an odor-specific affinity, so that its coverage by the odor is the product of concentration and affinity. Right, Olfactory neurons A and B have the same receptor type but different global sensitivities, neuron C has a different receptor type. Colored curves schematically represent the sensitivity to different odors, defined as the product of odor affinity and global sensitivity. Synchrony occurs at intersection points, for specific odors. D, More generally, a structured stimulus is described as the image of a lower-dimensional stimulus X through some transformation T. Synchrony occurs in two different neurons when their receptive fields match when combined with the transformation T.

Figure 8. Synchrony receptive fields in an olfactory model.

Top, An odor is presented with fluctuating concentration c(t). Receptor coverage is the affinity of the receptor type (type 1 for neuron C, type 2 for neurons A and B) times the concentration: a.c(t). The peak transduction current (middle) is a Hill function of receptor coverage, with different half-activation coverage for different neurons (inverse of global sensitivity). Neurons fire in synchrony to an odor when the product of odor affinity and global sensitivity match. This occurs for neurons B and C (black traces), but not for neurons A and C (dashed red trace).

Figure 9. Computing with synchrony in olfaction.

A, Top, Different odors produce different synchrony partitions (receptors with the same color are synchronous). Bottom, To each odor corresponds an assembly of postsynaptic neurons, where the inputs to each neuron belong to the same synchrony group (in each column, each postsynaptic with a given color receives synapses from all receptors with the same color). B, Top, Fluctuating concentration of three odors (A: blue, B: red, C: black). Middle, spiking responses of olfactory receptors. Bottom, Responses of postsynaptic neurons from the assembly selective to A (blue) and to B (red). Stimuli are presented is sequence: 1) odor A alone, 2) odor B alone, 3) odor B alone with twice stronger intensity, 4) odor A with distracting odor C (same intensity), 5) odors A and B (same intensity). C, Spike train statistics for the receptors (left column) and the postsynaptic neurons selective for odor A (right column), corresponding to the stimulation in the first 2 seconds of panel B. Top, distribution of firing rates; bottom, distribution of coefficients of variation. D, Top, Average firing rate in the assembly of postsynaptic neurons selective to A (blue) and in the assembly selective to B (red) when odor A is presented (as in panel B, first two seconds), as a function of the intrinsic noise (standard deviation relative to spike threshold). Bottom, Responses of the postsynaptic neurons for the maximum amount of intrinsic noise (σ = 0.5).

Figure 10. Recognition of odor mixtures and robustness.

A, Average firing rate of the postsynaptic assembly tuned to an equal mixture of odors A and B, as a function of the proportion of A in the presented mixture. Each curve corresponds to a different concentration (1, 10, 100). B, Binding: tuning curve of the postsynaptic assembly (same as in A for concentration 10) for mixtures presented in a single turbulent plume (solid) or in two independent plumes for the two odors (dashed). C, Same as in Fig. 6, but the membrane time constant of receptors is heterogeneous (between 15 and 25 ms). With the same synaptic projections as in Fig. 6 (initial wiring), the postsynaptic rate is reduced, but not odor specificity. The firing rate increases when the synaptic projections are adapted to this heterogeneity, i.e., presynaptic neurons have similar membrane time constants (new wiring).

Figure 11. Learning to detect odors.

A, Two odors are randomly presented to the network for 40 s. This histogram represents the distribution of tuning ratios after this learning period. The tuning ratio of a postsynaptic neuron is the proportion of spikes triggered by the first odor. B, Responses of postsynaptic neurons, ordered by tuning ratio, to odor A (blue) and odor B (red), with an increasing concentration (0.1 to 10, where 1 is odor concentration in the learning phase). C, Voltage traces for a postsynaptic neuron tuned to odor B, when odor A (left) and B (right) are presented.

Figure 12. Synchrony receptive fields in the auditory and visual modalities.

A, Binaural hearing with realistic sound diffraction. The sound S arrives at the two ears as a binaural signal (F_R*S, F_L*S), where F_R and F_L are location-dependent filters, and is subsequently processed by two monaural neurons with receptive fields N_A and N_B. The synchrony receptive field is the set of source locations such that N_A * F_R = N_B * F_L. B, Pitch. Two monaural neurons with different preferred frequencies fire in synchrony for a pure tone or resolved partial with frequency 1/f0, at the intersection of the two amplitude spectra (provided that the phase difference is compensated by appropriate delays). C, Binocular disparity. Two retinal ganglion cells fire in synchrony when there is an object at the convergence point of their fixation lines. D, Edges and textures. Two visual neurons with circular receptive fields fire in synchrony to images that are invariant to translations of the vector linking the two receptive field centers: edges with the same orientation and spatially periodic textures with the period given by that vector.