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Exact simulation of integrate-and-fire models with exponential currents

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Brette, R. (2007). Exact simulation of integrate-and-fire models with exponential currents. Neural Comput 19(10): 2604-2609.

Abstract. Neural networks can be simulated exactly using event-driven strategies, in which the algorithm advances directly from one spike to the next spike. It applies to neuron models for which we have 1) an explicit expression for the evolution of the state variables between spikes and 2) an explicit test on the state variables which predicts whether and when a spike will be emitted. In a previous work, we proposed a method which allows exact simulation of an integrate-and-fire model with exponential conductances, with the constraint of a single synaptic time constant. In this note we propose a method, based on polynomial root finding, which applies to integrate-and-fire models with exponential currents, with possibly many different synaptic time constants. Models can include biexponential synaptic currents and spike-triggered adaptation currents.

Download code:

  • Scilab implementation of functions for exact simulation and an example script for a random network: ScilabExpIF.zip. The archive includes an event-driven simulator written in Scilab for networks with random external events and without delays (this is for pedagogical purposes and is not intended to be efficient at all). N.B.: Scilab is a free scientific software (resembling Matlab).
  • C++ implementation of the same example: EventDrivenExpIF.zip (with the algorithm from the paper) and ClockDrivenExpIF.zip (with a standard clock-driven algorithm).

Neural development of binaural tuning through Hebbian learning predicts frequency-dependent best delays

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Fontaine, B. and Brette, R. (2011). Neural development of binaural tuning through Hebbian learning predicts frequency-dependent best delays. J Neurosci 31(32):11692–11696

Abstract. Birds use microsecond differences in the arrival times of the sounds at the two ears to infer the location of a sound source in the horizontal plane. These interaural time differences (ITDs) are encoded by binaural neurons which fire more when the ITD matches their "best delay". In the textbook model of sound localization, the best delays of binaural neurons reflect the differences in axonal delays of their monaural inputs, but recent observations have cast doubts on this classical view because best delays were found to depend on preferred frequency. Here we show that these observations are in fact consistent with the notion that best delays are created by differences in axonal delays, provided ITD tuning is created during development through spike-timing-dependent plasticity: basilar membrane filtering results in correlations between inputs to binaural neurons, which impact the selection of synapses during development, leading to the observed distribution of best delays.

Movie 1. Evolution of the synaptic weights of 3 neurons with CF = 2, 4 and 6, when presented with a binaurally delayed white noise with ITD = 167 µs.

Movie 2. Evolution of the best delays of 160 neurons with CF distributed between 2 kHz and 8 kHz, when presented with uncorrelated binaural noise.

Movie 3. Evolution of the best delays of 160 neurons with CF distributed between 2 kHz and 8 kHz, when presented with natural stereo recordings.

Spatial perception of pain (IV) Empirical evidence

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In this post I confront the propositions I previously described about where it hurts with experimental evidence. There is a recent review about spatial perception of pain, which contains a lot of relevant information (Haggard et al., 2013).

First of all, proposition A (spatial information is independently provided by tactile receptors) is quite clearly ruled out by empirical evidence. There are three types of nerve fibers innervating the skin. Aβ fibers mediate tactile sensations, while Aδ and C fibers mediate pain and temperature sensations. There is a lot of evidence that these are clearly separated, and the type of sensation does not depend on stimulation frequency (i.e., stimulating Aβ fibers never elicits pain). In addition, spatial localization does not seem to rely on Aβ fibers. For example, it is possible to block conduction in Aβ and Aδ fibers, leaving only C fibers. In this case, noxious stimuli can be localized on the skin with about the same resolution as when nerve conduction is normal (Koltzenburg et al., 1993). This implies that the pattern of activations of nociceptors conveys all the necessary information for spatial localization.

Now these patterns could be given a spatial meaning in different ways. One is learned association with tactile or visual stimuli (proposition C). In primary somatosensory cortex (S1), there are nociceptive somatotopic maps of single digits that are highly aligned with maps of responses to Aβ stimuli; there are also neurons that are sensitive to both mechanical stimulation and temperature. But this is only suggestive of a common spatial frame for both modalities. More specifically, if spatial information in pain-specific fibers were acquired from the tactile modality, then the spatial resolution of pain could never be better than that of touch – we would expect that it is similar and perhaps slightly worse. A systematic mapping on the whole body surface shows that this is the case in most locations, but not all (Mancini et al., 2014). Specifically, spatial resolution for pain is more accurate than for touch at the shoulder. In addition, the gradients of spatial resolution from shoulder to hand are opposite for touch and pain, in accordance with the gradients of innervation density of the corresponding fibers. Finally, there was one case of a subject completely lacking Aβ fibers that had normal spatial resolution. These observations rule out the proposition that spatial information in pain is acquired from the tactile modality (at least that it is entirely acquired from it).

It could be that spatial information in pain is acquired from vision. Although I have not seen such an experimental study, I would be very surprised to learn that blind people cannot localize pain. Finally, we could then postulate that spatial localization of pain is acquired from either touch or vision, whichever one is present. The best way to test it would be to map the spatial resolution of a blind subject lacking Aβ fibers. Without this test, the possibility is still quite implausible. Indeed, the subject lacking Aβ fibers had similar spatial resolution as other subjects. But the map includes the lower back, which cannot be seen directly (however the data is not shown for that specific location – I am assuming it follows the same pattern since the authors don't comment on it). Therefore, in that region there is neither vision nor touch for that subject. All these observations tend to reject proposition C.

There is an interesting observation about the relation between innervation density and spatial resolution. As I mentioned above, along the arm there are different gradients of spatial resolution between and touch and they agree with gradients of innervation density of the corresponding fibers. But in the fingertips, the relation does not hold: spatial resolution for pain is high but innervation density of pain-related fibers is low (Mancini et al., 2013). How is it possible? One possibility is cognitive factors, for example we use our hand a lot, perhaps attention is focused on the fingers or we have more experience grasping things and thus developing our spatial discrimination abilities. Another possibility (which I haven't seen mentioned in these studies) is that the patterns of activation may be intrinsically more discriminable, because of the shape, tissue composition or the presence of the fingerprints.

We are then left with proposition B (and variation B2): you feel pain at a particular location because specific movements that you make produce that pain. I noted earlier that this proposition raises a problem, which is that you cannot localize pain that you have not produced yourself in the past. It seems a bit implausible, when we think for example of tooth ache. I argued then that to solve this problem, one would need to postulate that nociceptors can be activated at a “subthreshold” level that does not produce pain. In this case, to feel pain at a particular location requires previously producing specific movements that produce a similar (but possibly less intense) pattern of activation of the pain receptors. The subthreshold activity of these fibers should reach the central nervous system and induce plastic changes supporting future localization of noxious stimuli, without producing any conscious sensation. Finally, I note that there is a potential problem in the fact that intensity and spatial information are carried through the same channel (pain-related fibers), and therefore intensity of pain changes the pattern of activation from which spatial information is extracted. If spatial localization is learned at subthreshold levels, then there is a potential issue of generalizing to pain-inducing levels, with possibilities for biases in pain localization.

Haggard P, Iannetti GD, Longo MR (2013) Spatial sensory organization and body representation in pain perception. Curr Biol CB 23:R164–176.
Koltzenburg M, Handwerker HO, Torebjörk HE (1993) The ability of humans to localise noxious stimuli. Neurosci Lett 150:219–222.
Mancini F, Bauleo A, Cole J, Lui F, Porro CA, Haggard P, Iannetti GD (2014) Whole-body mapping of spatial acuity for pain and touch. Ann Neurol 75:917–924.
Mancini F, Sambo CF, Ramirez JD, Bennett DLH, Haggard P, Iannetti GD (2013) A Fovea for Pain at the Fingertips. Curr Biol 23:496–500.

Late Emergence of the Whisker Direction Selectivity Map in the Rat Barrel Cortex

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Kremer Y, Léger JF, Goodman DF, Brette R, Bourdieu L (2011).Late Emergence of the Whisker Direction Selectivity Map in the Rat Barrel Cortex. J Neurosci 31(29):10689-700.

Abstract. In the neocortex, neuronal selectivities for multiple sensorimotor modalities are often distributed in topical maps thought to emerge during a restricted period in early postnatal development. Rodent barrel cortex contains a somatotopic map for whisker identity, but the existence of maps representing other tactile features has not been clearly demonstrated. We addressed the issue of the existence in the rat cortex of an intra-barrel map for whisker movement direction using in vivo two photon imaging. We discovered that the emergence of a direction map occurs long after all known critical periods in the somatosensory system. This map is remarkably specific, taking a pinwheel form centered near the barrel center and aligned to the barrel cortex somatotopy. We suggest that this map may arise from intra-cortical mechanisms and demonstrate by simulation that the combination of spike-timing-dependent plasticity at synapses between layer 4 and layer 2/3 and realistic pad stimulation is sufficient to produce such a map. Its late emergence long after other classical maps suggests that experience-dependent map formation and refinement continue throughout adult life.

Supplementary movie (20 MB). Spikes of L2/3 neurons and evolution of their direction selectivity during the stimulation with random moving bars. The movie shows the spikes of all neurons in L2/3 during the stimulation with random moving bars (left) and the evolution of their direction selectivity (right), as estimated from the average selectivity of their presynaptic neurons in L4, weighted by the synaptic weights. Initial conditions and parameters are identical to those used in the Figure 6. In these conditions, the simulation lasted 1500 seconds. Only 3% of all frames are included in the movie. Note that when a moving bar activates a sequence of whiskers, the first barrel column reached by the bar is almost homogeneously activated (at least at the beginning of the simulation) whereas subsequent barrel columns already exhibit inhomogeneous activity. This inhomogeneous activation of the second, third, etc barrel columns is not a consequence of the existence of a radial direction selectivity map, as this map does not have enough time to develop for instance at the first presentation of the moving bar. It is just due to horizontal connections within layer 2/3, which propagate the activation of the first column to the next adjacent columns, inhibiting predominantly the closest regions to the first column. This inhomogeneous activation of barrel columns becomes reinforced as the direction selectivity map develops, but this contribution only comes in second.

Code. In the examples of the Brian simulator, there is a script that stimulates a plastic model of the barrel cortex with randomly moving bars, and shows the selectivity map after learning.