Neural Computation via Neural Geometry: A Place Code for Inter-whisker Timing in the Barrel Cortex?
[['A fundamental question in computational neuroscience asks how the brain represents the relative timing of stimuli as they move between sensory receptors , e . g . as a light source moves relative to the retina , or as contact moves between touch sensors on the fingertip .', 'For over 60 years Jeffress’ place theory [1] has remained the dominant model .', 'The idea is that coincidence detector neurons receive input from sensors after delays governed by the distance of the neuron from either sensor .', 'The inter-sensor time difference is encoded by the location of neurons that are active because their connection delays exactly compensate the inter-sensor stimulation interval .', 'The place theory therefore suggests an important role for neural geometry in computing the motion of sensory stimuli .', 'Strong support for Jeffress’ place theory has been provided by a number of studies of midbrain neurons in auditory specialists like the barn owl , who locate sound sources by resolving small differences in the arrival time of sounds at either ear ( see ref . [2] for a review ) .', 'Evidence from the mammalian auditory system is less conclusive because , for example , rabbit auditory cortex neurons are tuned to inter-ear time differences that are too long to attribute to inter-neuron distances alone [3] ( see also refs . [4] , [5] , and ref . [6] for an alternative mechanism based on slow lateral connections ) .', 'However few studies have investigated how inter-sensor time-differences might be resolved in specialist mammalian sensory systems .', 'Tactile specialists like rats , mice , shrews , and seals determine the form and motion of tactile stimuli using prominent arrays of whiskers ( vibrissae ) on the face [7] , [8] .', 'For example , shrews hunting in the dark can use their whiskers to localise particular body-part shapes on fast-moving prey animals [9] .', 'Specific to the whisker system is a precise topographic correspondence between the individual sensor and its neural representation .', 'Deflection of adjacent whiskers A and B on the face evokes the largest amplitude and shortest latency responses in adjacent cortical columns A and B in the somatosensory ( barrel ) cortex .', 'This precise mapping , as well as observations of sub-millisecond temporal precision throughout [10]–[12] , makes the whisker-barrel system ideal for exploring the impact of neural geometry on neural computation .', 'A consistent finding across studies in the rat and mouse somatosensory cortex is that responses vary with the time interval between adjacent whisker stimulation [13]–[24] .', 'A useful metric for comparing the response to a two-whisker stimulus to the response to the individual whisker deflection is the facilitation index [17] , defined as ‘the response to paired deflection of whiskers A and B divided by the sum of the response to deflection of whisker A deflected alone and the response to whisker B deflected alone’ or .', 'In layer 2/3 barrel cortex ( L2/3 ) in particular , paired stimuli in which the adjacent whisker deflection precedes by 20– typically evoke sublinear responses ( ) .', 'For a range of near-simultaneous deflections , a number of studies have also reported supralinear responses ( ) , again particularly in L2/3 neurons [16]–[18] , [22] , [23] ( but see ref . [25] ) .', 'Interestingly Shimegi et al . [18] reported that septa-related neurons in L2/3 , located at the midline area between two barrels , were more likely to show response facilitation for short-interval stimuli , whereas barrel-related neurons were more likely to show response suppression by prior deflection of the distal whisker at longer intervals ( see Figure 1 ) .', 'Plots of the relationship between the inter-whisker-interval and the response magnitude for individual neurons showed evidence of tuning to particular short intervals .', 'Together these results suggest that the location of the L2/3 neuron relative to the underlying barrel geometry is important in determining its response to a two-whisker stimulus .', 'One explanation for the different responses of barrel-related and septa-related neurons , as summarised in Table 1 , is that they reflect the operation of different mechanisms for integrating adjacent-whisker signals in distinct barrel and septal circuits ( see refs . [26]–[28] ) .', 'However an alternative hypothesis , inspired by the place theory , is that the differences reflect an underlying continuum of responses , which are determined by the location of the neuron with respect to the two cortical columns .', 'This hypothesis would allow for , although it would not require , an essentially homogeneous population in L2/3 .', 'According to this alternative hypothesis , the relationship between the inter-whisker deflection interval and the facilitation index in L2/3 neurons may be determined by differences in the arrival times of synaptic inputs that originate from either barrel .', 'These differences may be attributed to inter-soma distance-dependent delays in the feed-forward projection from the major input in layer 4 barrel cortex ( L4 ) .', 'This hypothesis is supported by estimates of the speed of the projection between L4 and L2/3 neuron pairs that are relatively slow , around 0 . 2 meters per second for excitatory and inhibitory post-synaptic neurons [29] , [30] .', 'In this paper we show that simulated barrel cortex neurons that receive synaptic inputs with onset times constrained to embody this hypothesis can account for all of the trends relating to the stimulus interval in the data of ref .', '[18] .', 'We show that a natural prediction of the model is the existence of a topographic mapping of the inter-whisker deflection interval across the surface of L2/3 .', 'Specifically , supralinear population responses will peak at short non-zero intervals in neurons located closer to the barrel representing the later of the two deflected whiskers .', 'The responses of individual L2/3 neurons satisfy the basic requirements for a motion detector , and across the population these responses encode a range of stimulus motion velocities .', 'Results therefore suggest that two-whisker timing is represented by a place code in L2/3 barrel cortex .', 'More generally , the lateral displacement of active neurons due to distance-dependent delays on projections between cortical columns can be used to compute the sequence and timing of events between the sensory stimuli represented by activity in those columns .', 'The results are interpreted as evidence in support of the place theory as a general model of cortical processing of spatiotemporal information .'], ['We hypothesise that distance-dependent delays associated with inter-columar projections in sensory cortex can be used to extract the relative timing of sensory events .', 'Specifically , delays in the projection from layer 4 ( L4 ) to layer 2/3 ( L2/3 ) barrel cortex might generate selectivity to the inter-whisker deflection interval for adjacent whiskers .', 'To test the hypothesis , the latencies of synaptic inputs to a leaky integrate and fire neuron were constrained to reflect the range of geometries that characterise the L4 to L2/3 projection .', 'To validate the model , we recreated an adjacent-whisker paired-deflection study [18] , and compared responses of neurons in different cortical locations to stimuli in which the whiskers were deflected through a range of intervals .', 'The simplified model is based on three main assumptions , which are described with respect to the validation data in terms of adjacent whiskers A and B , but which in principle apply to a general model of cortical responses to arbitrarily complex multi-whisker deflection patterns .', 'The first assumption is that , upon whisker stimulation , inputs to L2/3 tend to originate from L4 neurons at the center of the corresponding barrel in L4 .', 'Therefore , in the model , the input layer L4 is collapsed down to just two point sources , with activity at each source representing the deflection of the corresponding whisker A or B . The second assumption is that the excitatory and inhibitory synaptic inputs evoked by deflection of whisker A and by deflection of whisker B arrive at a population of L2/3 neurons situated above and between corresponding barrels A and B . Therefore , in the model , each L2/3 neuron receives just four inputs , although each represents the total contribution of many similar synaptic contacts .', 'The third assumption is that the time taken for a L2/3 neuron to register a synaptic input is proportional to the straight-line distance between the L4 and L2/3 neuron .', 'Therefore , in the model , we assume that the time of arrival of each synaptic input is a linear function of the distance of the L2/3 neuron from either point source in L4 , and we refer to the associated constant of proportionality as the connection speed .', 'This simplified model of the neural geometry may deviate from the true situation .', 'For example , if the signalling delays are due to the axonal propagation speeds , then delays could be modified by the morphology of L4 axons , which branch vertically and laterally into L2/3 [31] , [32] .', 'Delays could also be modified by particular branching patterns that vary systematically with the location of the neuron in the home barrel [33] .', 'We choose not to explicitly model the variety of axonal morphologies , firstly to keep the model formulation simple , secondly because L4 to L2/3 signalling delays are well predicted by the straight-line inter-soma distance [29] , [30] , [34] , and thirdly because post-hoc simulations which considered a laterally-branching axonal morphology did not significantly alter the results .', 'Furthermore , recurrent interactions within L2/3 are not modelled explicitly , because they would occur subsequent to the initial activation of L2/3 , and thus could only affect the afferent response after the critical first spike response has been determined ( see Discussion ) .', 'Similarly , modelling each L4 input source as a discrete representation of one whisker is justified because multi-whisker responses in L4 are thought to be due to latent contributions from intra-cortical mechanisms [19] ( see Discussion ) .', 'The following sections outline how each assumption is represented formally in a model that we refer to as the distance-dependent delay hypothesis .', 'The plausibility of each assumption , the impact of each simplification , and the alternatives to each are considered in Discussion .', 'The thalamocortical volley of excitation from thalamus to L4 and then up into L2/3 [34] , [35] is closely followed by a volley of disynaptic inhibition , mediated by a small number of interneurons in L4 [36] , with a diverse range of morphologies [32] .', 'We posit that the main excitatory input to L2/3 is derived from direct synaptic connections from excitatory neurons in L4 , and the main inhibitory inputs are derived indirectly from excitation of L4 inhibitory interneurons .', 'The circuit therefore consists of three connections: an excitatory connection from L4 to L2/3 , an excitatory connection onto the L4 inhibitory interneuron , and an inhibitory connection from the L4 interneuron to the L2/3 neuron .', 'According to the distance-dependent delay hypothesis each connection has an associated delay .', 'The onset time of the direct excitatory synaptic input at the L2/3 neuron is proportional to its distance from the barrel center .', 'To model the indirect inhibition through an inhibitory interneuron we use a time delay proportional to the L4 to L2/3 distance plus a constant time delay accounting for the distance of the interneuron and its spike generation time .', 'The circuit therefore has three parameters: the speed of the excitatory pathway between L4 and the L2/3 target neuron ( ) , the speed of the inhibitory pathway between L4 and the L2/3 target neuron ( ) , and a fixed latency representing the delayed onset of the spike in the inhibitory interneuron ( ) relative to the onset of excitation in L4 .', 'For neurons in the barrel cortex , the principal whisker is typically defined as the one which , upon deflection , elicits the shortest latency and/or the largest-amplitude response .', 'Neurons of a particular barrel column tend to share the same principal whisker , the one which on the face is isomorphic with the position of the barrel in the grid of barrels .', 'For a given neuron all three criteria usually select the same whisker .', 'These constraints can be built into the model if , for progressively longer inter-soma distances , whisker-evoked inhibition arrives progressively earlier than excitation .', 'This pattern of delays requires that inhibitory connections are faster than excitatory connections , and that the onset of inhibition is delayed relative to the excitation .', 'This is achieved in the model by setting and .', 'In the analysis presented by Shimegi et al . [18] , against which the model will be validated , L2/3 neurons were characterised by their horizontal location with respect to two underlying barrel columns .', 'The geometry is shown in Figure 2 .', 'In the model axes and refer to orthogonal axes of the plane tangent to the pia matter of the brain ( i . e . , the plane tangential to the cortical surface ) [37]; specifically is aligned with barrels that correspond to a row of whiskers on the face , and is orthogonal in the ‘tangential plane’ .', 'The axis is normal to the tangential plane .', 'Axes and will henceforth be referred to as the horizontal and vertical axes respectively .', 'In the model , L2/3 neurons will be parameterised only by their horizontal location relative to the two input sources in L4 .', 'In effect , this means reducing the three spatial dimensions in which intra-cortical connections are defined to just two spatial dimensions by setting .', 'In this way we can define the position of two sources in L4 at .', 'Similarly we can describe L2/3 as a one-dimensional string and uniquely describe the location of individual L2/3 neurons along the string in terms of .', 'For example the neurons at , , and are L2/3 neurons located directly above barrel A , above barrel B , and above the midline respectively .', 'The Euclidean distance of each L2/3 neuron from the two sources can now be written in terms of : ( 1 ) ( 2 ) For the analyses presented in Results , the input sources were located at and the two layers were separated by vertical distance .', 'We will henceforth refer to and as inter-soma distances .', 'Reducing the description of the neural geometry in this way makes interpretation of the behaviour of the model tractable , and it allows for a direct comparison with the available electrophysiological data .', 'We note that using an alternative geometry has little impact on the main results , as considered in detail in Discussion .', 'The L2/3 neuron receives excitatory and inhibitory synaptic inputs from each stimulated whisker .', 'Thus , under two-whisker stimulation , the time of each input is given by: ( 3 ) ( 4 ) ( 5 ) ( 6 ) The inter-whisker interval ( ) is the time of deflection of whisker A , relative to whisker B , which is always deflected at time', '0 . Thus if whisker A was deflected before whisker B , if whisker B was deflected before whisker A , and if then the whiskers were deflected simultaneously .', 'The relationship between the inter-soma distance and the onset time of excitation and inhibition is illustrated in Figure 3A .', 'The connection speeds were chosen to be and , which are in the range of estimates derived from electrophysiological data [29] , [30] , but we note that similar analyses have estimated speeds as slow as [34] .', 'The constant was chosen to delay the onset of inhibition relative to excitation by for the neuron located closest to either barrel center , i . e . , .', 'With the inter-soma distance constrained by the geometry of Equations 1 and 2 , the input onset times , described by the linear functions in Figure 3A , become hyperbolic functions of the neuron location , as shown in Figure 3B .', 'The model neuron is a simple integrate and fire neuron with inputs in the form of excitatory and inhibitory post-synaptic conductance changes ( EPSCs and IPSCs ) .', 'Parameters followed those reported by Puccini et al . [38] as a guide for neurons in the barrel cortex .', 'The time course of each input , following its onset at time , , or , was modelled as a normalised difference of two exponentials: ( 7 ) The normalisation term , where , ensures that the potential peaks at', '1 . For excitatory synapses and simulating AMPA receptor channel opening [39] , and ensuring that excitatory inputs peak at .', 'For inhibitory synapses and as used by Puccini et al . [38] to model GABA receptor channel opening , peaking later than the EPSC at as seen in electrophysiological data ( e . g . , ref . [40] ) .', 'The maximum EPSC amplitude was and the maximum IPSC conductance amplitude was ( similar to ref . [38] ) .', 'The relative amplitude and time course of the excitatory and inhibitory post-synaptic currents are illustrated in Figure 3C .', 'For the L2/3 neuron we used a standard leaky integrate and fire neuron [41] , again with parameters guided by those from ref .', '[38]: ( 8 ) where the membrane time constant , the resting potential , the reversal potential for synapses of type inhibitory , and for excitatory synapses .', 'The leak conductance was and hence the membrane resistance .', 'Gaussian noise with standard deviation was added to the membrane potential at each time step .', 'Integration was by the forward Euler method ( ) .', 'When the membrane potential reached a spike was recorded , and the membrane potential was set to .'], ['To anticipate how a L2/3 neuron might respond to independent deflections of either whisker , we first determine when the onset times of the EPSC and IPSC evoked by deflection of that whisker will be coincident .', 'We derive the time of coincidence by setting the onset times to be equal and rearranging: ( 9 ) ( 10 ) Therefore we can determine that when and hence we would expect to see the largest responses to deflection of whisker A because the excitatory input precedes the inhibitory input .', 'To test this , neurons through the range of locations were stimulated by applying a deflection to either whisker A or whisker B in isolation .', 'Analogous to the experimental procedure of ref .', '[18] , each trial began prior to the onset of the first whisker deflection and ended after the onset of the second deflection .', 'Spike counts were calculated over this time window for the results of all simulations , however we note that spikes were precisely timed to the whisker stimuli and so this choice of time window is not critical for the behaviour of the model ( see Figure S1 ) .', 'The spike rate is shown as an average over 50 trials in Figure 4A to allow direct comparison with the results of ref .', '[18] , and averaged over 5000 trials for clarity in 4B .', 'As expected , neurons located closer to a particular barrel spike more often in response to deflection of the corresponding whisker .', 'As the distance of the neuron from either source increases , the excitatory and inhibitory inputs evoked by the corresponding neuron register at the neuron closer together in time and thus the window of opportunity in which the EPSC can cause a spike decreases .', 'At longer inter-soma distances , the IPSC precedes the EPSC , and effectively silences the neuron .', 'These observations agree with the notion of the principal whisker as that represented by the barrel closest to the neuron , and which evokes the shortest latency and largest amplitude response .', 'Figure 4 shows the linear sum of the response to independent deflection of both whiskers .', 'These values for the linear sum are later used to construct facilitation index scores from the average spike counts obtained in paired whisker-deflection trials .', 'For independent deflections of either whisker , we have seen that the spike rate is dictated by the sequence and relative timing of the synaptic inputs .', 'Responses to paired whisker deflection stimuli are more complex because they are dictated by four PSCs rather than two and also by the .', 'However similar analysis of the relative arrival times of PSCs can be used to anticipate these responses .', 'To this end it is useful to consider regions of the space of possible neuron location and inter-whisker deflection intervals ( henceforth – space , see Figure 5A ) that are delineated by different ordering of arrival times of the four PSCs .', 'These regions are delineated by loci representing coincident arrival of each possible pair amongst the four PSCs .', 'Equations 9–10 represent two such pairs .', 'As their solutions are not dependent on the , Equations 9–10 describe four loci , which when plotted are straight lines at constant values of that divide – space into five columns in Figure 5A .', 'Solutions for the other four pairs of PSCs can be written as functions of as follows: ( 11 ) ( 12 ) ( 13 ) ( 14 ) The solutions to Equations 11–14 are also plotted in Figure 5A , and they further divide the columns into ‘rows’ .', 'For each region of the graph we can use the equations to state the sequence of inputs for each synaptic pair .', 'This is done by setting all signs to signs in Equations 9–14 .', 'The eight inequalities that define each region of the graph can then be combined to give the order of all four synaptic PSCs , and the twenty-four possible PSC orderings take the form , for example , in the top-left region of – space shown in Figure 5A .', 'Considering now only whether each synaptic event in the input sequence is excitatory or inhibitory , we can describe the input to the L2/3 neuron more simply .', 'This effectively reduces the twenty-four PSC sequences to just six different orders in which excitation and inhibition can arrive at the neuron .', 'Figure 5B shows how each of the six orderings delineates a zone in – space .', 'For a range of short interval stimuli , neurons situated near the midline receive both excitatory inputs before both inhibitory inputs .', 'They receive inputs in the order , which can be read as ‘two excitations followed by two inhibitions’ .', 'This zone is coloured dark blue in Figure 5B .', 'It is in this zone that we would expect to observe the greatest spike rate because neither IPSC precedes the EPSCs .', 'Notice that this zone is oriented diagonally in – space , and therefore neurons in different locations near the midline will prefer a range of ( short ) .', 'Similarly we can expect that the greatest suppressive interactions will be displayed in the yellow ( ) , brown ( ) , and orange zones ( ) , in which an IPSC event is always registered first .', 'Of these zones the orange will be expected to yield the smallest suppression as the second IPSC is preceded by both EPSCs .', 'In the blue zone ( ) we might expect just one of the whisker deflections to evoke a response , as the second EPSC will be silenced by two preceding IPSCs .', 'In the cyan zones ( ) both EPSCs are followed immediately by an IPSC .', 'Therefore we might expect that if the two EPSC/IPSC pairs are separated sufficiently in time for the neuron to respond to them independently , i . e . , if the first inhibition has little effect on the second excitation , then the response will resemble the linear sum of that evoked by either whisker deflected independently , and hence the facilitation index score here will be around one .', 'Neurons through the range of locations were stimulated by applying paired deflections to whisker A and whisker B in sequence .', 'By analogy with the experimental procedure of ref .', '[18] , each trial began prior to the onset of the first whisker deflection and ended after the onset of the second .', 'The spike rate is shown as an average over 50 trials in Figure 5C .', 'As anticipated , the greatest activity was evoked in neurons around the midline ( ) when the whiskers were deflected through a range of short inter-whisker intervals ( ) .', 'Within this range neurons located left of the midline and therefore closer to barrel A responded maximally to slightly positive inter-whisker intervals where whisker B was deflected before whisker A . Neurons to the right of the midline and therefore closer to barrel B responded maximally when whisker A was deflected before whisker B at short intervals .', 'For intervals longer than around in either direction , and for neurons further from the midline than around half a millimetre , responses were much smaller .', 'In a region of – space roughly corresponding with the light blue zone in Figure 5B , responses were more variable at around 0 . 2 spikes per stimulus .', 'These results from the full spiking model fit well those expected based on the relative timing of the synaptic inputs .', 'Thus changing the relative timing of the synaptic inputs with distance-dependent delays alters the response of the neuron to paired whisker stimuli in a predictable way .', 'A major feature predicted by the simulation data is a mapping of short interval stimuli to the location of the most active L2/3 neuron .', 'The simulation data presented thus far suggest that distance-dependent delays in the L4 to L2/3 projection can generate a spatial encoding of the relative timing of whisker inputs for short interval stimuli .', 'But to what extent do these observations match up with experimental data ?', 'To answer this question we look first at the responses of individual model neurons to the range of different interval stimuli .', 'Figures 6A and 6B show the average spike rate for an individual neuron located either close to barrel B or between barrels A and B respectively .', 'The neuron in Figure 6A was located approximately to the right of the midline .', 'Also indicated in the figure is the linear sum of the response of this neuron to either whisker deflected in isolation .', 'Where paired stimuli evoke responses equal to this value , a facilitation index of 1 would be measured and we would conclude that no facilitatory interaction had occurred .', 'Where it is less , suppression would have been measured , and where it is greater facilitation would have been measured .', 'The neuron in Figure 6A shows no facilitatory interaction when whisker B ( the principal whisker ) is deflected prior to the adjacent whisker A . However for slightly negative intervals strong facilitation was measured , with the average spike count exceeding the linear sum baseline three-fold or more around a peak when whisker A is deflected before whisker B . When whisker A precedes by more than the response is strongly suppressed and almost no spikes are evoked .', 'The suppression recovers towards the linear sum baseline for intervals exceeding .', 'For the example midline neuron shown in Figure 6B facilitation appears more symmetrical around the zero inter-whisker interval .', 'Facilitation peaks for simultaneous intervals and fluctuates around baseline for longer intervals in either direction .', 'The peak in the average spike count is larger than that for the previous neuron , as is the linear sum response used to compute the strength of its facilitatory interaction .', 'Equivalent plots for individual L2/3 neurons , found in refs .', '[17] , [18] , [23] , [24] , display similar qualitative trends to those in Figure 6A and 6B , in terms of both the facilitatory interactions and of the average spike counts for independent and paired whisker stimuli .', 'In Figure 6C we group the L2/3 neurons by location as either above barrel A , above barrel B or in the septal region between the barrels .', 'This allows for a direct comparison between the simulation data ( Figure 6C ) and the available experimental data of ref .', '[18] ( compare with Figure 1 ) .', 'The simulation data share many of the qualities of the experimental data , as summarised in Table 1 .', 'Septal neurons show a large facilitatory peak for near simultaneous paired whisker deflections and for longer intervals in either direction respond with an average , equivalent to the response to either independently deflected whisker .', 'Neurons located above barrel B display on average a lesser facilitatory peak at interval stimuli , are suppressed by prior deflection of whisker B , and display no facilitatory interactions when whisker B is deflected first .', 'Geometry in the model is symmetrical about the midline and therefore the responses are symmetrical about the zero inter-whisker interval .', 'Therefore the above barrel B population display the exact opposite interactions with respect to the interval compared with the above barrel A population .', 'This includes a lesser peak for interval stimuli not apparent in the electrophysiological data .', 'Notice too that the peak of the septal group in the experimental data is for a slightly negative inter-whisker interval .', 'We will shortly demonstrate how an extension to the model , which introduces asymmetries related to the direction in which each whisker is deflected , may account for these differences .', 'For now we note that the population response predicted by the model affords a good match to the experimental data .', 'Instead of asking how L2/3 neurons in particular locations respond to different interval stimuli , we can ask how particular interval stimuli are represented across the population of L2/3 .', 'It is particularly important to consider the population response because even the most effective stimuli typically elicit less than one spike per stimulus in any particular neuron , and so individual spikes yield ambiguous information about the stimulus [42] .', 'Figure 7 shows the distribution of average responses across the population for a range of positive intervals .', 'Each of the short inter-whisker deflection intervals is clearly associated with a tuning curve across the population , with a peak that shifts to the left ( negative ) and scales systematically with the increase in interval .', 'Negative intervals also evoke symmetrical results , i . e . , a shift in peak responses towards neurons on the right , but we do not show them in the figure for clarity .', 'Viewed in this way , it is clear that the model predicts the existence of a topographic map for the inter-whisker deflection interval across the surface of L2/3 barrel cortex .', 'According to the model , paired whisker stimuli should elicit supralinear responses and display a systematic shift in tuning across the population for stimulus intervals ranging to .', 'As well as the representation of the inter-whisker interval across cortical space , it is useful to consider how the stimulus is represented in the timing of spikes .', 'Inspection of maps for the spike timing revealed that in paired-whisker stimulations , spikes were precisely timed to the whisker stimuli .', 'Moreover the largest responses reflected a combination of the delayed response to the principal whisker , as well as the superposition of excitatory influences from both whiskers ( see Figure S1 ) .', 'Therefore the model predicts that the effects measured by ref .', '[18] primarily operate on the first somatosensory-evoked spikes in L2/3 .', 'Barrel cortex neurons are selective for the direction in which the whiskers are deflected .', 'The mechanism thought to underlie directional selectivity in L4 neurons is similar to that which we have outlined for two-whisker timing , but with distances measured in degrees from the preferred stimulus direction [38] , [40] .', 'Several studies have suggested that direction preferences vary systematically within the barrel column , such that deflection of the principal whisker to the left or right is correlated with increased activity in neurons located to the equivalent left or right of the barrel column [43] , [44] .', 'Therefore we can model the effect of deflecting the whisker in either direction by moving the L4 point source for that whisker in either direction in L4 .', 'Accordingly , to represent a deflection of whisker A to the left ( away from whisker B ) we offset the point source in L4 that corresponds to whisker A by a fixed distance to obtain a new source location at .', 'Deflecting whisker A to the right means moving the point source to and similarly deflecting whisker B to the left or right means moving the second source to .', 'For two whiskers and two deflection directions , possible combinations are both deflections to the left ( leftwards ) , both right ( rightwards ) , A left & B right ( outwards ) , and A right & B left ( inwards ) .'
['Introduction', 'Materials and Methods', 'Results', 'Discussion']
['The place theory proposed by Jeffress ( 1948 ) is still the dominant model of how the brain represents the movement of sensory stimuli between sensory receptors .', 'According to the place theory , delays in signalling between neurons , dependent on the distances between them , compensate for time differences in the stimulation of sensory receptors .', 'Hence the location of neurons , activated by the coincident arrival of multiple signals , reports the stimulus movement velocity .', 'Despite its generality , most evidence for the place theory has been provided by studies of the auditory system of auditory specialists like the barn owl , but in the study of mammalian auditory systems the evidence is inconclusive .', 'We ask to what extent the somatosensory systems of tactile specialists like rats and mice use distance dependent delays between neurons to compute the motion of tactile stimuli between the facial whiskers ( or ‘vibrissae’ ) .', 'We present a model in which synaptic inputs evoked by whisker deflections arrive at neurons in layer 2/3 ( L2/3 ) somatosensory ‘barrel’ cortex at different times .', 'The timing of synaptic inputs to each neuron depends on its location relative to sources of input in layer 4 ( L4 ) that represent stimulation of each whisker .', 'Constrained by the geometry and timing of projections from L4 to L2/3 , the model can account for a range of experimentally measured responses to two-whisker stimuli .', 'Consistent with that data , responses of model neurons located between the barrels to paired stimulation of two whiskers are greater than the sum of the responses to either whisker input alone .', 'The model predicts that for neurons located closer to either barrel these supralinear responses are tuned for longer inter-whisker stimulation intervals , yielding a topographic map for the inter-whisker deflection interval across the surface of L2/3 .', 'This map constitutes a neural place code for the relative timing of sensory stimuli .']
['To perceive how stimuli move over sensor surfaces like the retina or the fingertips , neurons in the brain must report the relative timing of signals arriving at different locations on the sensor surface .', 'The rat whisker system is ideal for exploring how the brain performs this computation , because the layout of a small number of sensors ( whiskers ) maps directly onto the layout of corresponding columns of neurons in the sensory cortex .', 'Previous studies have found that neurons located between adjacent cortical columns are most likely to respond when the corresponding adjacent whiskers are stimulated in rapid succession .', 'These results suggest a link between the location of the neuron and the relative timing of sensory signals reported by its activity .', 'We hypothesized that , if the time taken for whisker signals to arrive at a neuron is related to its distance from each cortical column , then neurons closer to a particular column will report stimuli moving towards that particular whisker .', 'In a model approximating the geometry of cortical connections , responses of artificial neurons matched those of real neurons on a wide range of details .', 'These results suggest an important role for neural geometry in neural computation .']
['circuit models', 'auditory system', 'computational neuroscience', 'biology', 'computational biology', 'sensory systems', 'neuroscience', 'sensory perception', 'coding mechanisms']