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Showing papers on "Summation published in 2017"


Journal ArticleDOI
TL;DR: This work finds that all visual areas exhibit subadditive summation, whereby responses to longer stimuli are less than the linear prediction from briefer stimuli, and builds predictive models that operate on arbitrary temporal patterns of stimulation using two simple computations: temporal summation followed by a compressive nonlinearity.
Abstract: Combining sensory inputs over space and time is fundamental to vision. Population receptive field models have been successful in characterizing spatial encoding throughout the human visual pathways. A parallel question, how visual areas in the human brain process information distributed over time, has received less attention. One challenge is that the most widely used neuroimaging method, fMRI, has coarse temporal resolution compared with the time-scale of neural dynamics. Here, via carefully controlled temporally modulated stimuli, we show that information about temporal processing can be readily derived from fMRI signal amplitudes in male and female subjects. We find that all visual areas exhibit subadditive summation, whereby responses to longer stimuli are less than the linear prediction from briefer stimuli. We also find fMRI evidence that the neural response to two stimuli is reduced for brief interstimulus intervals (indicating adaptation). These effects are more pronounced in visual areas anterior to V1-V3. Finally, we develop a general model that shows how these effects can be captured with two simple operations: temporal summation followed by a compressive nonlinearity. This model operates for arbitrary temporal stimulation patterns and provides a simple and interpretable set of computations that can be used to characterize neural response properties across the visual hierarchy. Importantly, compressive temporal summation directly parallels earlier findings of compressive spatial summation in visual cortex describing responses to stimuli distributed across space. This indicates that, for space and time, cortex uses a similar processing strategy to achieve higher-level and increasingly invariant representations of the visual world.SIGNIFICANCE STATEMENT Combining sensory inputs over time is fundamental to seeing. Two important temporal phenomena are summation, the accumulation of sensory inputs over time, and adaptation, a response reduction for repeated or sustained stimuli. We investigated these phenomena in the human visual system using fMRI. We built predictive models that operate on arbitrary temporal patterns of stimulation using two simple computations: temporal summation followed by a compressive nonlinearity. Our new temporal compressive summation model captures (1) subadditive temporal summation, and (2) adaptation. We show that the model accounts for systematic differences in these phenomena across visual areas. Finally, we show that for space and time, the visual system uses a similar strategy to achieve increasingly invariant representations of the visual world.

70 citations


Journal ArticleDOI
TL;DR: Three types of experiments are presented, using neuronal cultures, indicating that each neuron functions as a collection of independent threshold units and that spatial summation and subtraction are not achieved when combining intra- and extra- cellular stimulations, as well as for nonlocal time interference, where the precise timings of the stimulations are irrelevant.
Abstract: Neurons are the computational elements that compose the brain and their fundamental principles of activity are known for decades. According to the long-lasting computational scheme, each neuron sums the incoming electrical signals via its dendrites and when the membrane potential reaches a certain threshold the neuron typically generates a spike to its axon. Here we present three types of experiments, using neuronal cultures, indicating that each neuron functions as a collection of independent threshold units. The neuron is anisotropically activated following the origin of the arriving signals to the membrane, via its dendritic trees. The first type of experiments demonstrates that a single neuron’s spike waveform typically varies as a function of the stimulation location. The second type reveals that spatial summation is absent for extracellular stimulations from different directions. The third type indicates that spatial summation and subtraction are not achieved when combining intra- and extra- cellular stimulations, as well as for nonlocal time interference, where the precise timings of the stimulations are irrelevant. Results call to re-examine neuronal functionalities beyond the traditional framework, and the advanced computational capabilities and dynamical properties of such complex systems.

65 citations


Journal ArticleDOI
05 Dec 2017-eLife
TL;DR: It is shown that serotonin (5HT), which is known to regulate gamma power, acts via 5HT2A receptors to suppress an inward-rectifying potassium conductance in FSIs, leading to depolarization, increased input resistance, enhanced spiking, and slowed decay of excitatory post-synaptic potentials (EPSPs).
Abstract: The medial prefrontal cortex plays a key role in higher order cognitive functions like decision making and social cognition. These complex behaviors emerge from the coordinated firing of prefrontal neurons. Fast-spiking interneurons (FSIs) control the timing of excitatory neuron firing via somatic inhibition and generate gamma (30-100 Hz) oscillations. Therefore, factors that regulate how FSIs respond to gamma-frequency input could affect both prefrontal circuit activity and behavior. Here, we show that serotonin (5HT), which is known to regulate gamma power, acts via 5HT2A receptors to suppress an inward-rectifying potassium conductance in FSIs. This leads to depolarization, increased input resistance, enhanced spiking, and slowed decay of excitatory post-synaptic potentials (EPSPs). Notably, we found that slowed EPSP decay preferentially enhanced temporal summation and firing elicited by gamma frequency inputs. These findings show how changes in passive membrane properties can affect not only neuronal excitability but also the temporal filtering of synaptic inputs.

35 citations


Journal ArticleDOI
TL;DR: Results indicate that SM produces specific inhibitory effects on temporal summation of back pain, consistent with the involvement of a spinal anti-nociceptive mechanism in clinical pain relief by SM, providing the first mechanistic evidence of backpain relief by spinal manipulation.

29 citations


Journal ArticleDOI
01 Jun 2017-Pain
TL;DR: It is indicated that lateral inhibition limits SSP and is an intrinsic component of nociceptive information processing, which may contribute substantially to the radiation of some types of chronic pain.
Abstract: Spatial summation of pain (SSP) is the increase of perceived intensity that occurs as the stimulated area increases. Spatial summation of pain is subadditive in that increasing the stimulus area produces a disproportionately small increase in the perceived intensity of pain. A possible explanation for subadditive summation may be that convergent excitatory information is modulated by lateral inhibition. To test the hypothesis that lateral inhibition may limit SSP, we delivered different patterns of noxious thermal stimuli to the abdomens of 15 subjects using a computer-controlled CO2 laser. Lines (5 mm wide) of variable lengths (4, 8 cm) were compared with 2-point stimuli delivered at the same position/separation as the length of lines. When compared with one-point control stimuli, 2-point stimulus patterns produced statistically significant SSP, while no such summation was detected during line stimulus patterns. Direct comparison of pain intensity evoked by 2-point pattern stimuli with line pattern stimuli revealed that 2-point patterns were perceived as significantly more painful, despite the fact that the 2-point pattern stimulated far smaller areas of skin. Thus, the stimulation of the skin region between the endpoints of the lines appears to produce inhibition. These findings indicate that lateral inhibition limits SSP and is an intrinsic component of nociceptive information processing. Disruption of such lateral inhibition may contribute substantially to the radiation of some types of chronic pain.

29 citations


Journal ArticleDOI
TL;DR: The relatively long integration window measured in this study may be relevant to studies of the absolute visual threshold, i.e., tests of single‐photon vision, where “single” photons should be separated by greater than the integration window to avoid summation.

19 citations


Posted ContentDOI
14 Sep 2017-bioRxiv
TL;DR: This work finds that all visual areas exhibit sub-additive summation, whereby responses to longer stimuli are less than the linear prediction from briefer stimuli, and develops a general model that shows how these effects can be captured with two simple operations: temporal summation followed by a compressive nonlinearity.
Abstract: Combining sensory inputs over space and time is fundamental to vision. Population receptive field models have been highly successful in characterizing spatial encoding throughout the human visual pathways. A parallel question-how visual areas in the human brain process information distributed over time-has received less attention. One challenge is that the most widely used neuroimaging method-fMRI-has coarse temporal resolution compared to the time scale of neural dynamics. Here, via carefully controlled temporally modulated stimuli, we show that information about temporal processing can be readily derived from fMRI signal amplitudes. We find that all visual areas exhibit subadditive summation, whereby responses to longer stimuli are less than the linear prediction from briefer stimuli. We also find fMRI evidence that the neural response to two stimuli is reduced for brief interstimulus intervals (indicating adaptation). These effects are more pronounced in anterior visual areas than V1-V3. Finally, we develop a general model that shows how all of these effects can be captured with two simple operations: temporal summation followed by a compressive nonlinearity. This model operates for arbitrary temporal stimulation patterns and provides a simple and interpretable set of computations that can be used to characterize neural response properties across the visual hierarchy. Importantly, compressive temporal summation directly parallels earlier findings of compressive spatial summation in human visual cortex describing responses to stimuli distributed across space. This indicates that for space and time, cortex uses a similar processing strategy to achieve higher-level and increasingly invariant representations of the visual world.

19 citations


Journal ArticleDOI
TL;DR: Results show that radiate multipolar neurons integrate a large number of weak synaptic inputs over a broad dynamic range, and have intrinsic and synaptic properties that are distinct from planar multipolar neuron.
Abstract: Radiate and planar neurons are the two major types of multipolar neurons in the ventral cochlear nucleus (VCN). Both cell types receive monosynaptic excitatory synaptic inputs from the auditory nerve, but have different responses to sound and project to different target regions and cells. Although the intrinsic physiology and synaptic inputs to planar neurons have been previously characterized, the radiate neurons are less common and have not been as well studied. We studied both types of multipolar neurons and characterized their properties including intrinsic excitability, synaptic dynamics of their auditory nerve inputs, as well as their neural firing properties to auditory nerve stimulation. Radiate neurons had a faster member time constant and higher threshold current to fire spikes than planar neurons, but the maximal firing rate is the same for both cell types upon large current injections. Compared to planar neurons, radiate neurons showed spontaneous postsynaptic currents with smaller size, and slower but variable kinetics. Auditory nerve stimulation progressively recruited synaptic inputs that were smaller and slower in radiate neurons, over a broader range of stimulus strength. Synaptic inputs to radiate neurons showed less depression than planar neurons during low rates of repetitive activity, but the synaptic depression at higher rates was similar between two cell types. However, due to the slow kinetics of the synaptic inputs, synaptic transmission in radiate neurons showed prominent temporal summation that contributed to greater synaptic depolarization and a higher firing rate for repetitive auditory nerve stimulation at high rates. Taken together, these results show that radiate multipolar neurons integrate a large number of weak synaptic inputs over a broad dynamic range, and have intrinsic and synaptic properties that are distinct from planar multipolar neurons. These properties enable radiate neurons to generate powerful inhibitory inputs to target neurons during high levels of afferent activity. Such robust inhibition is expected to dynamically modulate the excitability of many cell types in the cochlear nuclear complex.

19 citations


Journal ArticleDOI
TL;DR: Direct evidence is presented suggesting the constant contact, ramp and hold stimulus may underestimate the level of TSSP, and the re-evaluation of stimulation techniques used for temporal summation tests, especially within clinical models are suggested.

16 citations


Journal ArticleDOI
TL;DR: This study sought to provide a preliminary assessment of whether spinally mediated afferent hyperactivity might contribute to manifestations of overactive bladder syndrome (OAB) in women as indexed by elevated temporal summation of evoked heat pain stimuli.
Abstract: Introduction This study sought to provide a preliminary assessment of whether spinally mediated afferent hyperactivity (i.e., central sensitization) might contribute to manifestations of overactive bladder syndrome (OAB) in women as indexed by elevated temporal summation of evoked heat pain stimuli. Methods We recruited 20 adult women with OAB who were planning to undergo interventional therapy for OAB with either onabotulinumtoxinA injection or sacral neuromodulation and 23 healthy controls without OAB symptoms to undergo quantitative sensory testing with cutaneous thermal pain temporal summation. The primary study outcome was the degree of temporal summation, as reflected in the magnitude of positive slope of the line fitted to the series of 10 stimuli at the 49°C target temperatures. Linear regression and analysis of covariance were utilized to compare the degree of temporal summation between study groups. Results The standardized slope of temporal summation trials for women with OAB was significantly higher than for controls (β = 3.43, 95% confidence interval = 0.6–6.2, P = 0.017). The adjusted means ±SE of the standardized temporal summation slopes for the full OAB and control groups were 3.0 ± 0.5 (95% confidence interval = 2.0, 4.1) and 1.7 ± 0.5 (95% confidence interval = 0.7, 2.7), respectively. Conclusion In this preliminary study, we demonstrated that women with OAB refractory to primary and secondary therapies exhibited greater thermal cutaneous temporal summation than women without OAB symptoms. This suggests that central sensitization, indexed by temporal summation, may be an underlying factor contributing to OAB in some women. Neurourol. Urodynam. © 2016 Wiley Periodicals, Inc.

13 citations


Journal ArticleDOI
TL;DR: This work has shown that in humans, temporal summation of the nociceptive withdrawal reflex of the lower limb is an objective representation of the temporal processing of nOCiceptive signals into the spinal cord.

Journal ArticleDOI
TL;DR: The results indicate that D1 receptor activation enhances the temporal summation of R GCs mainly by suppressing Kir currents through the cAMP/PKA signaling pathway, thus increasing the excitability of rat RGCs.

Journal ArticleDOI
TL;DR: The hypothesis that chickens use spatial summation of cone signals to maintain color vision in low light levels is tested by determining the intensity thresholds of color discrimination using similar stimuli, patterns of grey tiles of varying intensity interspersed with color tiles, adjusted for this specific aim.

Journal ArticleDOI
TL;DR: This study demonstrated that the properties of area- and distance-based summation are uniform across the neck, back, and forearm in healthy adults, refuting the thesis that spatial summation of pain may be a contributing factor for the reported difference in chronicity rates between spinal and peripheral sites.

Journal ArticleDOI
TL;DR: The results show that two types of muscle reflex activity are differentially modulated 1 mo after spinal cord injury (SCI) and that spinal reflexes are altered in a muscle-specific manner during this critical period.
Abstract: This research is the first to assess temporal summation, also called wind-up, of muscle reflexes during the 1-mo recovery period following spinal injury. Our results show that two types of muscle r...

Posted ContentDOI
22 Mar 2017-bioRxiv
TL;DR: A stochastic model is developed that incorporates noise mechanisms in synaptic transmission, and it is argued that probabilistic release of neurotransmitters is not only a source of disturbance, but plays a beneficial role in synaptic information processing.
Abstract: In the nervous system, communication occurs via synaptic transmission where signaling molecules (neurotransmitters) are released by the presynaptic neuron, and they influence electrical activity of another neuron (postsynaptic neuron). The inherent probabilistic release of neurotransmitters is a significant source of noise that critically impacts the timing of spikes (action potential) in the postsynaptic neuron. We develop a stochastic model that incorporates noise mechanisms in synaptic transmission, such as, random docking of neurotransmitter-filled vesicle to a finite number of docking sites, with each site having a probability of vesicle release upon arrival of an action potential. This random, burst-like release of neurotransmitters serves as an input to an integrate-and-fire model, where spikes in the postsynaptic neuron are triggered when its membrane potential reaches a critical threshold for the first time. We derive novel analytical results for the probability distribution function of spike timing, and systematically investigate how underlying model parameters and noise processes regulate variability in the inter-spike times. Interestingly, in some parameter regimes, independent arrivals of action potentials in the presynaptic neuron generate strong dependencies in the spike timing of the postsynaptic neuron. Finally, we argue that probabilistic release of neurotransmitters is not only a source of disturbance, but plays a beneficial role in synaptic information processing.


Journal ArticleDOI
TL;DR: In this paper, a dual-emitter light-induced neuromorphic device composed of two lightinduced devices with a common collector and base was proposed, where two InGaN multiple quantum well diodes (MQW-diodes) were used as the emitters to generate light, and one In-GaN MQW-diode was used as a collector to absorb the emitted light.
Abstract: We propose and fabricate a dual-emitter light-induced neuromorphic device composed of two light-induced devices with a common collector and base. Two InGaN multiple quantum well diodes (MQW-diodes) are used as the emitters to generate light, and one InGaN MQW-diode is used as the common collector to absorb the emitted light. When the presynaptic voltages are synchronously applied to the two emitters, the collector demonstrates an adding together of the excitatory post synaptic voltage (EPSV). The width and period of the two input signals constitute the code to generate spatial summation and recognition effects at the same time. Experimental results confirm that temporal summation caused by the repetitive-pulse facilitation could significantly strengthen the spatial summation effect due to the adding together behavior when the repetitive stimulations are applied to the two emitters in rapid succession. Particularly, the resonant summation effect occurs at the co-summation region when the two repetitive-pulse signals have a resonant period, which offers a more sophisticated spatiotemporal EPSV summation function for the dual-emitter neuromorphic device.

Journal ArticleDOI
TL;DR: It is found that an L-type Ca2+ channel antagonist, nifedipine, applied at concentrations of 50 μM or 100 μM to the hindlimb enlargement spinal cord, significantly reduced windup of flexion reflex motor patterns, while lower concentrations of nifEDipine had no effect.
Abstract: Windup is a form of multisecond temporal summation in which identical stimuli, delivered seconds apart, trigger increasingly strong neuronal responses. L-type Ca2+ channels have been shown to play an important role in the production of windup of spinal cord neuronal responses, initially in studies of turtle spinal cord and later in studies of mammalian spinal cord. L-type Ca2+ channels have also been shown to contribute to windup of limb withdrawal reflex (flexion reflex) in rats, but flexion reflex windup has not previously been described in turtles and its cellular mechanisms have not been studied. We studied windup of flexion reflex motor patterns, evoked with weak mechanical and electrical stimulation of the dorsal hindlimb foot skin and assessed via a hip flexor (HF) nerve recording, in spinal cord-transected and immobilized turtles in vivo. We found that an L-type Ca2+ channel antagonist, nifedipine, applied at concentrations of 50 μM or 100 μM to the hindlimb enlargement spinal cord, significantly reduced windup of flexion reflex motor patterns, while lower concentrations of nifedipine had no such effect. Nifedipine similarly reduced the amplitude of an individual flexion reflex motor pattern evoked by a stronger mechanical stimulus, in a dose-dependent manner, suggesting that L-type Ca2+ channels contribute to each flexion reflex as well as to multisecond summation of flexion reflex responses in turtles. We also found that we could elicit flexion reflex windup consistently using a 4-g von Frey filament, which is not usually considered a nociceptive stimulus. Thus, it may be that windup can be evoked by a wide range of tactile stimuli and that L-type calcium channels contribute to multisecond temporal summation of diverse tactile stimuli across vertebrates.

Journal ArticleDOI
TL;DR: Direct evidence is reported in two neurological patients for neural summation in the superior colliculus, and that it is mediated by afferent visual information transmitted through its brachium, providing the first evidence that visual afferents to the SC contribute to the efficiency of target detection.
Abstract: Two visual signals appearing simultaneously are detected more rapidly than either signal appearing alone. Part of this redundant target effect (RTE) can be attributed to neural summation that has been proposed to occur in the superior colliculus (SC). We report direct evidence in two neurological patients for neural summation in the SC, and that it is mediated by afferent visual information transmitted through its brachium. The RTE was abolished in one patient with a hemorrhage involving the right posterior thalamus that damaged part of the SC and that disrupted its brachium; and in another patient in whom the SC appeared intact but deafferented due to traumatic avulsion of its brachium. In addition reaction time for unilateral targets in the contralesional field was slowed in both patients, providing the first evidence that visual afferents to the SC contribute to the efficiency of target detection.

Posted ContentDOI
14 Feb 2017-bioRxiv
TL;DR: It is proposed that several features of temporal responses - adaptation, summation, and the timescale of temporal dynamics - can be understood as resulting from a small number of canonical neuronal computations.
Abstract: The visual system analyzes image properties across multiple spatial and temporal scales. Population receptive field ("pRF") models have successfully characterized spatial representations across the human visual pathways. Here, we studied temporal representations, measuring fMRI and electrocorticographic ("ECoG") responses in posterior, lateral, ventral, and dorsal visual areas to briefly viewed contrast patterns. We built a temporal pRF model employing linear summation and time-varying divisive normalization. Our model accurately predicts the fMRI amplitude and ECoG broadband time-course, accounting for two phenomena - accumulation of stimulus information over time (summation), and response reduction with prolonged or repeated exposure (adaptation). We find systematic differences in these properties: summation periods are increasingly long and adaptation more pronounced in higher compared to earlier visual areas. We propose that several features of temporal responses - adaptation, summation, and the timescale of temporal dynamics - can be understood as resulting from a small number of canonical neuronal computations.

Posted Content
TL;DR: A computational model of neuron, called firing cell (FC), properties of which cover such phenomena as attenuation of receptors for external stimuli, delay and decay of postsynaptic potentials, modification of internal weights due to propagation of post synapses through the dendrite are proposed.
Abstract: We propose a computational model of neuron, called firing cell (FC), properties of which cover such phenomena as attenuation of receptors for external stimuli, delay and decay of postsynaptic potentials, modification of internal weights due to propagation of postsynaptic potentials through the dendrite, modification of properties of the analog memory for each input due to a pattern of short-time synaptic potentiation or long-time synaptic potentiation (LTP), output-spike generation when the sum of all inputs exceeds a threshold, and refraction. The cell may take one of the three forms: excitatory, inhibitory, and receptory. The computer simulations showed that, depending on the phase of input signals, the artificial neuron's output frequency may demonstrate various chaotic behaviors.

Journal ArticleDOI
TL;DR: How the modulation of the number of inputs and/or the frequency firing can explain the orientation selectivity in V1 and three different hypotheses namely the spatial summation, the temporal summation and the excitation-inhibition are proposed.
Abstract: Neurons in the primary visual cortex (V1) are sensitive to simple features of the visual scene such as contrast, spatial frequency or orientations. In higher mammals, they are organized into columns of orientation-preference, whereas such organization is absent in rodents. However, in both types of organization, neurons can be highly selective or poorly selective for a particular stimulus. In mouse V1, it has been shown that all stimuli are represented on the dendritic tree of single neurons. To what extent this concept is applicable in higher mammals? In this review, we discuss possible models of integrating visual information from visual cortical neurons. In particular, how the modulation of the number of inputs and/or the frequency firing can explain the orientation selectivity in V1. Based on our findings and literature, we propose three different hypotheses namely the spatial summation, the temporal summation and the excitation-inhibition. In addition, we discuss the possible interactions between excitatory pyramidal neurons and inhibitory interneurons during stimulus processing.

Journal ArticleDOI
TL;DR: This study evaluated the time‐dependent behaviour of mechanical characteristics of soft tissue during repeated cuff stimulation used for eliciting temporal summation of cuff pressure‐evoked pain.

Journal ArticleDOI
TL;DR: Responses of multiple target muscles diverge across a physiological activity range as a result of distinct synaptic properties sensitive to multiple time scales, as measured in muscles innervated by a chewing circuit neuron.
Abstract: Adaptive changes in the output of neural circuits underlying rhythmic behaviors are relayed to muscles via motor neuron activity. Pre- and postsynaptic properties of neuromuscular junctions can impact the transformation from motor neuron activity to muscle response. Further, synaptic plasticity occurring on the time scale of inter-spike intervals can differ between multiple muscles innervated by the same motor neuron. In rhythmic behaviors, motor neuron bursts can elicit additional synaptic plasticity. However, it is unknown if plasticity regulated by the longer time scale of inter-burst intervals also differs between synapses from the same neuron, and whether any such distinctions occur across a physiological activity range. To address these issues, we measured electrical responses in muscles innervated by a chewing circuit neuron, the lateral gastric (LG) motor neuron, in a well-characterized small motor system, the stomatogastric nervous system (STNS) of the Jonah crab, Cancer borealis . In vitro and in vivo, sensory, hormonal and modulatory inputs elicit LG bursting consisting of inter-spike intervals of 50-250 ms and inter-burst intervals of 2-24 s. Muscles expressed similar facilitation measured with paired stimuli except at the shortest inter-spike interval. However distinct decay time constants resulted in differences in temporal summation. In response to bursting activity, augmentation occurred to different extents and saturated at different inter-burst intervals in the three muscles. Further, augmentation interacted with facilitation, resulting in distinct intra-burst facilitation between muscles. Thus, responses of multiple target muscles diverge across a physiological activity range due to distinct synaptic properties sensitive to multiple time scales.

Posted ContentDOI
29 Mar 2017-bioRxiv
TL;DR: It is proposed that several features of temporal responses - adaptation, summation, and the timescale of temporal dynamics - can be understood as resulting from a small number of canonical neuronal computations.
Abstract: The visual system analyzes image properties across multiple spatial and temporal scales. Population receptive field ("pRF") models have successfully characterized spatial representations across the human visual pathways. Here, we studied temporal representations, measuring fMRI and electrocorticographic ("ECoG") responses in posterior, lateral, ventral, and dorsal visual areas to briefly viewed contrast patterns. We built a temporal pRF model employing linear summation and time-varying divisive normalization. Our model accurately predicts the fMRI amplitude and ECoG broadband time-course, accounting for two phenomena -accumulation of stimulus information over time (summation), and response reduction with prolonged or repeated exposure (adaptation). We find systematic differences in these properties: summation periods are increasingly long and adaptation more pronounced in higher compared to earlier visual areas. We propose that several features of temporal responses - adaptation, summation, and the timescale of temporal dynamics - can be understood as resulting from a small number of canonical neuronal computations.