From Wikipedia, the free encyclopedia

Neural coding is a neuroscience-related field concerned with characterizing the relationship between the stimulus and the individual or ensemble neuronal responses and the relationship among electrical activity (biology) of the neurons in the ensemble.^[1] Based on the theory that sensory and other information is represented in the brain by networks ofneurons, it is thought that neurons can encode both digital and analog information^[2] (see Neural networks).

Overview[edit]

Neurons are remarkable among the cells of the body in their ability to propagate signals rapidly over large distances. They do this by generating characteristic electrical pulses called action potentials or, more simply, spikes that can travel down nerve fibers. Sensory neurons change their activities by firing sequences of action potentials in various temporal patterns, with the presence of external sensory stimuli, such as light, sound, taste, smell and touch. It is known that information about the stimulus is encoded in this pattern of action potentials and transmitted into and around the brain.

Although action potentials can vary somewhat in duration, amplitude and shape, they are typically treated as identical stereotyped events in neural coding studies. If the brief duration of an action potential (about 1ms) is ignored, an action potential sequence, or spike train, can be characterized simply by a series of all-or-none point events in time.^[3] The lengths of interspike intervals (ISIs) between two successive spikes in a spike train often vary, apparently randomly.^[4] The study of neural coding involves measuring and characterizing how stimulus attributes, such as light or sound intensity, or motor actions, such as the direction of an arm movement, are represented by neuron action potentials or spikes. In order to describe and analyze neuronal firing, statistical methods and methods of probability theory and stochastic point processes have been widely applied.

Encoding and decoding[edit]

The link between stimulus and response can be studied from two opposite points of view. Neural encoding refers to the map from stimulus to response. The main focus is to understand how neurons respond to a wide variety of stimuli, and to construct models that attempt to predict responses to other stimuli. Neural decoding refers to the reverse map, from response to stimulus, and the challenge is to reconstruct a stimulus, or certain aspects of that stimulus, from the spike sequences it evokes.

Coding schemes[edit]

A sequence, or 'train', of spikes may contain information based on different coding schemes. In motor neurons, for example, the strength at which an innervated muscle is flexed depends solely on the 'firing rate', the average number of spikes per unit time (a 'rate code'). At the other end, a complex 'temporal code' is based on the precise timing of single spikes. They may be locked to an external stimulus such as in the auditory system or be generated intrinsically by the neural circuitry.^[5]

Whether neurons use rate coding or temporal coding is a topic of intense debate within the neuroscience community, even though there is no clear definition of what these terms mean.

Rate coding[edit]

Rate coding is a traditional coding scheme, assuming that most, if not all, information about the stimulus is contained in the firing rate of the neuron. Because the sequence of action potentials generated by a given stimulus varies from trial to trial, neuronal responses are typically treated statistically or probabilistically. They may be characterized by firing rates, rather than as specific spike sequences. In most sensory systems, the firing rate increases, generally non-linearly, with increasing stimulus intensity.^[6] Any information possibly encoded in the temporal structure of the spike train is ignored. Consequently, rate coding is inefficient but highly robust with respect to the ISI 'noise'.^[4]

The concept of firing rates has been successfully applied during the last 80 years. It dates back to the pioneering work of ED Adrian who showed that the firing rate of stretchreceptor neurons in the muscles is related to the force applied to the muscle.^[7] In the following decades, measurement of firing rates became a standard tool for describing the properties of all types of sensory or cortical neurons, partly due to the relative ease of measuring rates experimentally. However, this approach neglects all the information possibly contained in the exact timing of the spikes. During recent years, more and more experimental evidences have suggested that a straightforward firing rate concept based on temporal averaging may be too simplistic to describe brain activity.^[4]

During rate coding, precisely calculating firing rate is very important. In fact, the term "firing rate" has a few different definitions, which refer to different averaging procedures, such as an average over time or an average over several repetitions of experiment.

In rate coding, learning is based on activity-dependent synaptic weight modifications.

Spike-count rate[edit]

The Spike-count rate, also referred to as temporal average, is obtained by counting the number of spikes that appear during a trial and dividing by the duration of trial. The length T of the time window is set by experimenter and depends on the type of neuron recorded from and the stimulus. In practice, to get sensible averages, several spikes should occur within the time window. Typical values are T = 100 ms or T = 500 ms, but the duration may also be longer or shorter.

The spike-count rate can be determined from a single trial, but at the expense of losing all temporal resolution about variations in neural response during the course of the trial. Temporal averaging can work well in cases where the stimulus is constant or slowly varying and does not require a fast reaction of the organism — and this is the situation usually encountered in experimental protocols. Real-world input, however, is hardly stationary, but often changing on a fast time scale. For example, even when viewing a static image, humans perform saccades, rapid changes of the direction of gaze. The image projected onto the retinal photoreceptors changes therefore every few hundred milliseconds.

Despite its shortcomings, the concept of a spike-count rate code is widely used not only in experiments, but also in models of neural networks. It has led to the idea that a neuron transforms information about a single input variable (the stimulus strength) into a single continuous output variable (the firing rate).

Time-dependent firing rate[edit]

The time-dependent firing rate is defined as the average number of spikes (averaged over trials) appearing during a short interval between times t and t+Δt, divided by the duration of the interval. It works for stationary as well as for time-dependent stimuli. To experimentally measure the time-dependent firing rate, the experimenter records from a neuron while stimulating with some input sequence. The same stimulation sequence is repeated several times and the neuronal response is reported in a Peri-Stimulus-Time Histogram (PSTH). The time t is measured with respect to the start of the stimulation sequence. The Δt must be large enough (typically in the range of one or a few milliseconds) so there are sufficient number of spikes within the interval to obtain a reliable estimate of the average. The number of occurrences of spikes n_K(t;t+Δt) summed over all repetitions of the experiment divided by the number K of repetitions is a measure of the typical activity of the neuron between time t and t+Δt. A further division by the interval length Δt yields time-dependent firing rate r(t) of the neuron, which is equivalent to the spike density of PSTH.

For sufficiently small Δt, r(t)Δt is the average number of spikes occurring between times t and t+Δt over multiple trials. If Δt is small, there will never be more than one spike within the interval between t and t+Δt on any given trial. This means that r(t)Δt is also the fraction of trials on which a spike occurred between those times. Equivalently, r(t)Δt is theprobability that a spike occurs during this time interval.

As an experimental procedure, the time-dependent firing rate measure is a useful method to evaluate neuronal activity, in particular in the case of time-dependent stimuli. The obvious problem with this approach is that it can not be the coding scheme used by neurons in the brain. Neurons can not wait for the stimuli to repeatedly present in an exactly same manner before generating response.

Nevertheless, the experimental time-dependent firing rate measure can make sense, if there are large populations of independent neurons that receive the same stimulus. Instead of recording from a population of N neurons in a single run, it is experimentally easier to record from a single neuron and average over N repeated runs. Thus, the time-dependent firing rate coding relies on the implicit assumption that there are always populations of neurons.

Temporal coding[edit]

When precise spike timing or high-frequency firing-rate fluctuations are found to carry information, the neural code is often identified as a temporal code.^[8] A number of studies have found that the temporal resolution of the neural code is on a millisecond time scale, indicating that precise spike timing is a significant element in neural coding.^[2][9]

Temporal codes employ those features of the spiking activity that cannot be described by the firing rate. For example, time to first spike after the stimulus onset, characteristics based on the second and higher statistical moments of the ISI probability distribution, spike randomness, or precisely timed groups of spikes (temporal patterns) are candidates for temporal codes.^[10] As there is no absolute time reference in the nervous system, the information is carried either in terms of the relative timing of spikes in a population of neurons or with respect to an ongoing brain oscillation.^[2][4]

The temporal structure of a spike train or firing rate evoked by a stimulus is determined both by the dynamics of the stimulus and by the nature of the neural encoding process. Stimuli that change rapidly tend to generate precisely timed spikes and rapidly changing firing rates no matter what neural coding strategy is being used. Temporal coding refers to temporal precision in the response that does not arise solely from the dynamics of the stimulus, but that nevertheless relates to properties of the stimulus. The interplay between stimulus and encoding dynamics makes the identification of a temporal code difficult.

The issue of temporal coding is distinct and independent from the issue of independent-spike coding. If each spike is independent of all the other spikes in the train, the temporal character of the neural code is determined by the behavior of time-dependent firing rate r(t). If r(t) varies slowly with time, the code is typically called a rate code, and if it varies rapidly, the code is called temporal.

Phase-of-firing code is a type of code which takes into account a time label for each spike according to a time reference based on phase of local ongoing oscillations at low^[11] or high frequencies.^[12] A feature of this code is that neurons adhere to a preferred order of spiking, resulting in firing sequence.^[13]

In temporal coding, learning can be explained by activity-dependent synaptic delay modifications.^[14] The modifications can themselves depend not only on spike rates (rate coding) but also on spike timing patterns (temporal coding), i.e., can be a special case of spike-timing-dependent plasticity.

Population coding[edit]

Population coding is a method to represent stimuli by using the joint activities of a number of neurons. In population coding, each neuron has a distribution of responses over some set of inputs, and the responses of many neurons may be combined to determine some value about the inputs.

From the theoretical point of view, population coding is one of a few mathematically well-formulated problems in neuroscience. It grasps the essential features of neural coding and yet, is simple enough for theoretic analysis.^[15] Experimental studies have revealed that this coding paradigm is widely used in the sensor and motor areas of the brain. For example, in the visual area medial temporal (MT), neurons are tuned to the moving direction.^[16] In response to an object moving in a particular direction, many neurons in MT fire, with a noise-corrupted and bell-shaped activity pattern across the population. The moving direction of the object is retrieved from the population activity, to be immune from the fluctuation existing in a single neuron’s signal.

Population coding has a number of advantages, including reduction of uncertainty due to neuronal variability and the ability to represent a number of different stimulus attributes simultaneously. Population coding is also much faster than rate coding and can reflect changes in the stimulus conditions nearly instantaneously.^[17] Individual neurons in such a population typically have different but overlapping selectivities, so that many neurons, but not necessarily all, respond to a given stimulus.

Position coding[edit]

A typical population code involves neurons with a Gaussian tuning curve whose means vary linearly with the stimulus intensity, meaning that the neuron responds most strongly (in terms of spikes per second) to a stimulus near the mean. The actual intensity could be recovered as the stimulus level corresponding to the mean of the neuron with the greatest response. However, the noise inherent in neural responses means that a maximum likelihood estimation function is more accurate.

This type of code is used to encode continuous variables such as joint position, eye position, color, or sound frequency. Any individual neuron is too noisy to faithfully encode the variable using rate coding, but an entire population ensures greater fidelity and precision. For a population of unimodal tuning curves, i.e. with a single peak, the precision typically scales linearly with the number of neurons. Hence, for half the precision, twice as many neurons are required. In contrast, when the tuning curves have multiple peaks, as in grid cells that represent space, the precision of the population can scale exponentially with the number of neurons. This greatly reduces the number of neurons required for the same precision.^[18]

References[edit]

1. ^ Brown EN, Kass RE, Mitra PP (May 2004). "Multiple neural spike train data analysis: state-of-the-art and future challenges". Nat. Neurosci. 7 (5): 456–61.doi:10.1038/nn1228. PMID 15114358.
2. ^ ^{a b c} Thorpe, S.J. (1990). "Spike arrival times: A highly efficient coding scheme for neural networks". In Eckmiller, R.; Hartmann, G.; Hauske, G. Parallel processing in neural systems and computers (PDF). North-Holland. pp. 91–94. ISBN 978-0-444-88390-2.
3. ^ Gerstner, Wulfram; Kistler, Werner M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press. ISBN 978-0-521-89079-3.
4. ^ ^{a b c d} Stein RB, Gossen ER, Jones KE (May 2005). "Neuronal variability: noise or part of the signal?". Nat. Rev. Neurosci. 6 (5): 389–97. doi:10.1038/nrn1668.PMID 15861181.
5. ^ Gerstner W, Kreiter AK, Markram H, Herz AV (November 1997). "Neural codes: firing rates and beyond". Proc. Natl. Acad. Sci. U.S.A. 94 (24): 12740–1. PMC 34168.PMID 9398065.
6. ^ Kandel, E.; Schwartz, J.; Jessel, T.M. (1991). Principles of Neural Science (3rd ed.). Elsevier. ISBN 0444015620.
7. ^ Adrian ED, Zotterman Y (April 1926). "The impulses produced by sensory nerve-endings: Part II. The response of a Single End-Organ". J. Physiol. (Lond.) 61 (2): 151–71. PMC 1514782. PMID 16993780.
8. ^ Dayan, Peter; Abbott, L. F. (2001). Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. Massachusetts Institute of Technology Press. ISBN 978-0-262-04199-7.
9. ^ Butts DA, Weng C, Jin J, et al. (September 2007). "Temporal precision in the neural code and the timescales of natural vision". Nature 449 (7158): 92–5.doi:10.1038/nature06105. PMID 17805296.
10. ^ Kostal L, Lansky P, Rospars JP (November 2007). "Neuronal coding and spiking randomness". Eur. J. Neurosci. 26 (10): 2693–701. doi:10.1111/j.1460-9568.2007.05880.x. PMID 18001270.
11. ^ Montemurro MA, Rasch MJ, Murayama Y, Logothetis NK, Panzeri S (March 2008)."Phase-of-firing coding of natural visual stimuli in primary visual cortex". Curr. Biol. 18(5): 375–80. doi:10.1016/j.cub.2008.02.023. PMID 18328702.
12. ^ Fries P, Nikolić D, Singer W (July 2007). "The gamma cycle". Trends Neurosci. 30(7): 309–16. doi:10.1016/j.tins.2007.05.005. PMID 17555828.
13. ^ Havenith MN, Yu S, Biederlack J, Chen NH, Singer W, Nikolić D (June 2011)."Synchrony makes neurons fire in sequence, and stimulus properties determine who is ahead". J. Neurosci. 31 (23): 8570–84. doi:10.1523/JNEUROSCI.2817-10.2011.PMID 21653861.
14. ^ Geoffrois, E.; Edeline, J.M.; Vibert, J.F. (1994). "Learning by Delay Modifications". In Eeckman, Frank H. Computation in Neurons and Neural Systems. Springer. pp. 133–8.ISBN 978-0-7923-9465-5.
15. ^ Wu S, Amari S, Nakahara H (May 2002). "Population coding and decoding in a neural field: a computational study". Neural Comput 14 (5): 999–1026.doi:10.1162/089976602753633367. PMID 11972905.
16. ^ Maunsell JH, Van Essen DC (May 1983). "Functional properties of neurons in middle temporal visual area of the macaque monkey. I. Selectivity for stimulus direction, speed, and orientation". J. Neurophysiol. 49 (5): 1127–47. PMID 6864242.
17. ^ Hubel DH, Wiesel TN (October 1959). "Receptive fields of single neurones in the cat's striate cortex". J. Physiol. (Lond.) 148: 574–91. PMC 1363130. PMID 14403679.
18. ^ Mathis A, Herz AV, Stemmler MB (July 2012). "Resolution of nested neuronal representations can be exponential in the number of neurons". Phys. Rev. Lett. 109(1): 018103. PMID 23031134.

From Wikipedia, the free encyclopedia

Neural decoding is a neuroscience-related field concerned with the reconstruction of sensory and other stimuli from information that has already been encoded and represented in the brain by networks of neurons. Reconstruction refers to the ability of the researcher to predict what sensory stimuli the subject is receiving based purely on neuron action potentials. Therefore, the main goal of neural decoding is to characterize how the electrical activity of neurons elicit activity and responses in the brain.^[1]

This article specifically refers to neural decoding as it pertains to the mammalian neocortex.

Overview[edit]

When looking at a picture, our brains are constantly making decisions about what object we are looking at, where we need to move our eyes next, and what we find to be the most salient aspects of the input stimulus. As these images hit the back of our retina, these stimuli are converted from varying wavelengths to a series of neural spikes called action potentials. These pattern of action potentials are different for different objects and different colors; we therefore say that the neurons are encoding objects and colors by varying their spike rates or temporal pattern. Now, if someone were to probe the brain by placing electrodes in the primary visual cortex, they may find what appears to be random electrical activity. These neurons are actually firing in response to the lower level features of visual input, possibly the edges of a picture frame. This highlights the crux of the neural decoding hypothesis: that is possible to reconstruct a stimulus from the response of the ensemble of neurons that represent it. By this we mean, it is possible to look at spike train data and say that the person or animal we are recording from is looking at a red ball.

Encoding to decoding[edit]

Implicit about the decoding hypothesis is the assumption that neural spiking in the brain somehow represents stimuli in the external world. The decoding of neural data would be impossible if the neurons were firing randomly: nothing would be represented. This process of decoding neural data forms a loop with neural encoding. First, the organism must be able to perceive a set of stimuli in the world - say a picture of a hat. Seeing the stimuli must result in some internal learning: the encoding stage. After varying the range of stimuli that is presented to the observer, we expect the neurons to adapt to the statistical properties of the signals, encoding those that occur most frequently:^[2] the efficient-coding hypothesis. Now neural decoding is the process of taking these statistical consistencies, a statistical model of the world, and reproducing the stimuli. This may map to the process of thinking and acting, which in turn guide what stimuli we receive, and thus, completing the loop.

In order to build a model of neural spike data, one must both understand how information is originally stored in the brain and how this information is used at a later point in time. Thisneural coding and decoding loop is a symbiotic relationship and the crux of the brain's learning algorithm. Furthermore, the processes that underlie neural decoding and encoding are very tightly coupled and may lead to varying levels of representative ability^[3][4]

Spatial resolutions[edit]

Much of the neural decoding problem depends on the spatial resolution of the data being collected. The goal here is to answer the question: how many neurons do I need to record in order to reconstruct the stimulus with reasonable accuracy. This question intimately relates to the means by which data is collected as it relates to the area being recorded. Neurons with small areas of coverage such as rods and cones in the retina may require more recordings than simple cells in the primary visual cortex. Here, rods and cones only respond to the color of small visual area, while simple cells respond to the orientation of lines.

Previous recording methods relied on stimulating single neurons over a repeated series of tests in order to generalize this neuron's behavior.^[5] New techniques such as high-densitymulti-electrode array recordings and multi-photon calcium imaging techniques now make it possible record from upwards of a few hundred neurons. Even with better recording techniques, the focus of these recordings must be on an area of the brain that is both manageable and qualitatively understood. Many studies look at spike train data gathered from the ganglion cells in the retina. Of all the possible subset of neurons to study, this particular area has the benefits of being strictly feedforward, retinotopic, and amenable to current recording granularities. The duration, intensity, and location of the stimulus can be controlled guaranteeing that a predetermined number of ganglion cells can be sampled within a significantly structured microcosm of the visual system.^[6] In addition to the visual system, other studies evaluate the discriminatory ability of rat facial whiskers^[7] and the olfactory coding of moth pheromone receptor neurons^[8] as mediums for collected spike train data.

Even with ever-improving recording techniques, one will always run into the limited sampling problem: given a limited number of recording trials, it is impossible to completely account for the error associated with noisy data obtained from stochastically functioning neurons (for example, a neuron's electric potential fluctuates around its resting potentialdue to a constant influx and efflux of sodium and potassium ions). Therefore, it is not possible to perfectly reconstruct a stimulus from spike data. Luckily, even with noisy data, the stimulus can still be reconstructed within acceptable error bounds.^[9]

Temporal resolutions[edit]

Another important consideration to take into account when decoding the neural code are the timescales and frequencies of the stimulus being presented to the observer. Quicker timescales and higher frequencies demand faster and more precise responses in neural spike data. In humans, millisecond precision has been observed throughout the visual cortex, the retina,^[10] and the lateral geniculate nucleus, so one would suspect this to be the appropriate measuring frequency. This has been confirmed in studies that quantify the responses of neurons in the lateral geniculate nucleus to white-noise and naturalistic movie stimuli.^[11] At the cellular level, spike-timing-dependent plasticity operates at millisecond timescales;^[12] therefore, models seeking biological relevance should be able to perform at these temporal scales.

Probabilistic decoding[edit]

When decoding neural data, arrival times of each spike

, and the probability of seeing a certain stimulus, may be the extent of the available data. The prior distribution

defines an ensemble of signals, and represents the likelihood of seeing a stimulus in the world based on previous experience. The spike times may also be drawn from a distribution

; however, what we want to know is the probability distribution over a set of stimuli given a series of spike trains , which is called the response-conditional ensemble. What remains is the characterization of the neural code by translating stimuli into spikes,

; the traditional approach to calculating this probability distribution has been to fix the stimulus and examine the responses of the neuron. Combining everything using Bayes' Ruleresults in the simplified probabilistic characterization of neural decoding:

. An area of active research consists of finding better ways of representing and determining

.^[13] The following are some such examples.

Spike train number[edit]

The simplest coding strategy is the spike train number coding. This method assumes that the spike number is the most important quantification of spike train data. In spike train number coding, each stimulus in represented by a unique firing rate across the sampled neurons. The color red may be signified by 5 total spikes across the entire set of neurons, while the color green may be 10 spikes; each spike is pooled together into an overall count. This is represented by:

the number of spikes, is the number of spikes of neuron at stimulus presentation time , and s is the stimulus.

Instantaneous rate code[edit]

Adding a small temporal component results in the spike timing coding strategy. Here, the main quantity measured is the number of spikes that occur within a predefined window of time T. This method adds another dimension to the previous. This timing code is given by:

where

is the jth spike on the lth presentation of neuron i, is the firing rate of neuron i at time t, and 0 to T is the start to stop times of each trial.

Temporal correlation[edit]

Temporal correlation code, as the name states, adds correlations between individual spikes. This means that the time between a spike

and its preceding spike is included. This is given by:

where

is the time interval between a neurons and the one preceding it.

Ising decoder[edit]

Another description of neural spike train data uses the Ising model borrowed from the physics of magnetic spins. Because neural spike trains effectively binarized(either on or off) at small time scales (10 to 20 ms), the Ising model is able to effectively capture the present pairwise correlations,^[14] and is given by:

where

is the set of binary responses of neuron i, is the external fields function, is the pairwise couplings function, and is the partition function[disambiguation needed].

Agent-based decoding[edit]

In addition to the probabilistic approach, agent-based models exist that capture the spatial dynamics of the neural system under scrutiny. One such model is hierarchical temporal memory, which is a machine learning framework that organizes visual perception problem into a hierarchy of interacting nodes (neurons). The connections between nodes on the same levels and a lower levels are termed synapses, and their interactions are subsequently learning. Synapse strengths modulate learning and are altered based on the temporal and spatial firing of nodes in response to input patterns.^[15][16]

While it is possible to take the firing rates of these modeled neurons, and transform them into the probabilistic and mathematical frameworks described above, agent-based models provide the ability to observe the behavior of the entire population of modeled neurons. Researchers can circumvent the limitations implicit with lab-based recording techniques. Because this approach does rely on modeling biological systems, error arises in the assumptions made by the researcher and in the data used in parameter estimation.

Applicability[edit]

The advancement in our understanding of neural decoding benefits the development of brain-machine interfaces, prosthetics^[17] and the understanding of neurological disorders such as epilepsy.^[18]

References[edit]

1. ^ Jacobs AL, Fridman G, Douglas RM, et al. (April 2009). "Ruling out and ruling in neural codes". Proc. Natl. Acad. Sci. U.S.A. 106 (14): 5936–41. doi:10.1073/pnas.0900573106.PMC 2657589. PMID 19297621.
2. ^ Barlow, H. (1961). Possible principles underlying the transformation of sensory messages. Sensory communication.
3. ^ Chacron MJ, Longtin A, Maler L (2004). "To burst or not to burst?". J Comput Neurosci 17 (2): 127–36. doi:10.1023/B:JCNS.0000037677.58916.6b. PMID 15306735.
4. ^ Boloori AR, Jenks RA, Desbordes G, Stanley GB (July 2010). "Encoding and decoding cortical representations of tactile features in the vibrissa system". J. Neurosci. 30 (30): 9990–10005. doi:10.1523/JNEUROSCI.0807-10.2010. PMC 2957657. PMID 20668184.
5. ^ Hubel DH, Wiesel TN, LeVay S (April 1977). "Plasticity of ocular dominance columns in monkey striate cortex". Philos. Trans. R. Soc. Lond., B, Biol. Sci. 278 (961): 377–409.PMID 19791.
6. ^ Warland DK, Reinagel P, Meister M (November 1997). "Decoding visual information from a population of retinal ganglion cells". J. Neurophysiol. 78 (5): 2336–50. PMID 9356386.
7. ^ Arabzadeh E, Panzeri S, Diamond ME (September 2006). "Deciphering the spike train of a sensory neuron: counts and temporal patterns in the rat whisker pathway". J. Neurosci. 26(36): 9216–26. doi:10.1523/JNEUROSCI.1491-06.2006. PMID 16957078.
8. ^ Kostal L, Lansky P, Rospars JP (April 2008). "Efficient olfactory coding in the pheromone receptor neuron of a moth". PLoS Comput. Biol. 4 (4): e1000053.doi:10.1371/journal.pcbi.1000053. PMC 2291565. PMID 18437217.
9. ^ Rolls ET, Treves A (November 2011). "The neuronal encoding of information in the brain". Prog. Neurobiol. 95 (3): 448–90. doi:10.1016/j.pneurobio.2011.08.002.PMID 21907758.
10. ^ Berry MJ, Meister M (March 1998). "Refractoriness and neural precision". J. Neurosci. 18 (6): 2200–11. PMID 9482804.
11. ^ Butts DA, Weng C, Jin J, et al. (September 2007). "Temporal precision in the neural code and the timescales of natural vision". Nature 449 (7158): 92–5. doi:10.1038/nature06105.PMID 17805296.
12. ^ Song S, Miller KD, Abbott LF (September 2000). "Competitive Hebbian learning through spike-timing-dependent synaptic plasticity". Nat. Neurosci. 3 (9): 919–26.doi:10.1038/78829. PMID 10966623.
13. ^ Rieke, F. (1999). Spikes: exploring the neural code. exploring the neural code (p. 395). The MIT Press.
14. ^ Schaub MT, Schultz SR (February 2012). "The Ising decoder: reading out the activity of large neural ensembles". J Comput Neurosci 32 (1): 101–18. doi:10.1007/s10827-011-0342-z. PMID 21667155.
15. ^ Hawkins, J., Ahmad, S., & Dubinsky, D. (2006). Hierarchical temporal memory: Concepts, theory and terminology. Whitepaper.
16. ^ Hawkins, J., & Blakeslee, S. (2005). On intelligence. Owl Books.
17. ^ Donoghue JP (November 2002). "Connecting cortex to machines: recent advances in brain interfaces". Nat. Neurosci. 5 (Suppl): 1085–8. doi:10.1038/nn947. PMID 12403992.
18. ^ Rolston JD, Desai SA, Laxpati NG, Gross RE (October 2011). "Electrical stimulation for epilepsy: experimental approaches". Neurosurg. Clin. N. Am. 22 (4): 425–42, v.doi:10.1016/j.nec.2011.07.010. PMC 3190668. PMID 21939841.