Hierarchical matrix

2013-11-29T19:37:17Z

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'''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 [[neural network|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 potential]]s. Therefore, the main goal of neural decoding is to characterize how the electrical activity of neurons elicit activity and responses in the brain.<ref name="Jacobs">{{cite journal |author=Jacobs AL, Fridman G, Douglas RM, ''et al.'' |title=Ruling out and ruling in neural codes |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=106 |issue=14 |pages=5936–41 |year=2009 |month=April |pmid=19297621 |pmc=2657589 |doi=10.1073/pnas.0900573106 |url=http://www.pnas.org/cgi/pmidlookup?view=long&pmid=19297621}}</ref>

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

== Overview ==
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 potential]]s. 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 [[visual cortex|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 ==
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 coding|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 [[Neural code|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 [[Signal processing|signals]], encoding those that occur most frequently:<ref name="barlow">Barlow, H. (1961). Possible principles underlying the transformation of sensory messages. Sensory communication.</ref> the [[Efficient coding hypothesis|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. This [[neural 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<ref name="Chacron">{{cite journal |author=Chacron MJ, Longtin A, Maler L |title=To burst or not to burst? |journal=J Comput Neurosci |volume=17 |issue=2 |pages=127–36 |year=2004 |pmid=15306735 |doi=10.1023/B:JCNS.0000037677.58916.6b }}</ref><ref name="Baloori">{{cite journal |author=Boloori AR, Jenks RA, Desbordes G, Stanley GB |title=Encoding and decoding cortical representations of tactile features in the vibrissa system |journal=J. Neurosci. |volume=30 |issue=30 |pages=9990–10005 |year=2010 |month=July |pmid=20668184 |pmc=2957657 |doi=10.1523/JNEUROSCI.0807-10.2010 |url=http://www.jneurosci.org/cgi/pmidlookup?view=long&pmid=20668184}}</ref>

== Spatial resolutions ==
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 cell]]s 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 [[Neurostimulation|stimulating single neurons]] over a repeated series of tests in order to generalize this neuron's behavior.<ref name="Hubel">{{cite journal |author=Hubel DH, Wiesel TN, LeVay S |title=Plasticity of ocular dominance columns in monkey striate cortex |journal=Philos. Trans. R. Soc. Lond., B, Biol. Sci. |volume=278 |issue=961 |pages=377–409 |year=1977 |month=April |pmid=19791 |url=http://rstb.royalsocietypublishing.org/cgi/pmidlookup?view=long&pmid=19791}}</ref> New techniques such as high-density [[Multielectrode array|multi-electrode array recordings]] and [[Two-photon excitation microscopy|multi-photon calcium imaging techniques]] now make it possible to 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 [[Retinal ganglion cell|ganglion cells]] in the retina. Of all the possible subset of neurons to study, this particular area has the benefits of being strictly [[Feedforward neural network|feedforward]], [[Retinotopy|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.<ref name="Warland">{{cite journal |author=Warland DK, Reinagel P, Meister M |title=Decoding visual information from a population of retinal ganglion cells |journal=J. Neurophysiol. |volume=78 |issue=5 |pages=2336–50 |year=1997 |month=November |pmid=9356386 |url=http://jn.physiology.org/cgi/pmidlookup?view=long&pmid=9356386}}</ref> In addition to the visual system, other studies evaluate the discriminatory ability of rat facial whiskers<ref name="Arabzadeh">{{cite journal |author=Arabzadeh E, Panzeri S, Diamond ME |title=Deciphering the spike train of a sensory neuron: counts and temporal patterns in the rat whisker pathway |journal=J. Neurosci. |volume=26 |issue=36 |pages=9216–26 |year=2006 |month=September |pmid=16957078 |doi=10.1523/JNEUROSCI.1491-06.2006 |url=http://www.jneurosci.org/cgi/pmidlookup?view=long&pmid=16957078}}</ref> and the olfactory coding of moth pheromone receptor neurons<ref name="kostal">{{cite journal |author=Kostal L, Lansky P, Rospars JP |title=Efficient olfactory coding in the pheromone receptor neuron of a moth |journal=PLoS Comput. Biol. |volume=4 |issue=4 |pages=e1000053 |year=2008 |month=April |pmid=18437217 |pmc=2291565 |doi=10.1371/journal.pcbi.1000053 |url=http://dx.plos.org/10.1371/journal.pcbi.1000053}}</ref> 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 potential]] due to a constant influx and efflux of [[Voltage-gated sodium channel|sodium]] and [[Voltage-gated potassium channel|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.<ref name="Rolls">{{cite journal |author=Rolls ET, Treves A |title=The neuronal encoding of information in the brain |journal=Prog. Neurobiol. |volume=95 |issue=3 |pages=448–90 |year=2011 |month=November |pmid=21907758 |doi=10.1016/j.pneurobio.2011.08.002 |url=http://linkinghub.elsevier.com/retrieve/pii/S0301-0082(11)00147-X}}</ref>

== Temporal resolutions ==
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]],<ref name="berry">{{cite journal |author=Berry MJ, Meister M |title=Refractoriness and neural precision |journal=J. Neurosci. |volume=18 |issue=6 |pages=2200–11 |year=1998 |month=March |pmid=9482804 |url=http://www.jneurosci.org/cgi/pmidlookup?view=long&pmid=9482804}}</ref> 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.<ref name="Butts">{{cite journal |author=Butts DA, Weng C, Jin J, ''et al.'' |title=Temporal precision in the neural code and the timescales of natural vision |journal=Nature |volume=449 |issue=7158 |pages=92–5 |year=2007 |month=September |pmid=17805296 |doi=10.1038/nature06105 }}</ref> At the cellular level, [[spike-timing-dependent plasticity]] operates at millisecond timescales;<ref name="Song">{{cite journal |author=Song S, Miller KD, Abbott LF |title=Competitive Hebbian learning through spike-timing-dependent synaptic plasticity |journal=Nat. Neurosci. |volume=3 |issue=9 |pages=919–26 |year=2000 |month=September |pmid=10966623 |doi=10.1038/78829 }}</ref> therefore, models seeking biological relevance should be able to perform at these temporal scales.

== Probabilistic decoding ==
When decoding neural data, arrival times of each spike <math>t_1,\text{ }t_2,\text{ }...,\text{ }t_n\text{ }=\text{ }\{t_i\}</math>, and the [[probability]] of seeing a certain stimulus, <math>P[s(t)]</math> may be the extent of the available data. The [[Prior probability|prior distribution]] <math>P[s(t)]</math> defines an ensemble of signals, and represents the [[Likelihood function|likelihood]] of seeing a stimulus in the world based on previous experience. The spike times may also be drawn from a [[Probability distribution|distribution]] <math>P[\{t_i\}]</math>; however, what we want to know is the [[probability distribution]] over a set of stimuli given a series of spike trains <math>P[s(t)|\{t_i\}]</math>, which is called the [[Conditional probability|response-conditional]] ensemble. What remains is the characterization of the neural code by translating stimuli into spikes, <math>P[\{t_i\}|s(t)]</math>; 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' Rule]] results in the simplified probabilistic characterization of neural decoding: <math>P[ s(t) | \{ t_i \} ] = P[ \{ t_i \} | s(t) ] * (P[s(t)]/P[\{t_i\}] )</math>. An area of active research consists of finding better ways of representing and determining <math>P[ \{ t_i \} | s(t) ]</math>.<ref name="Rieke">Rieke, F. (1999). Spikes: exploring the neural code. exploring the neural code (p. 395). The MIT Press.</ref> The following are some such examples.

=== Spike train number ===
The simplest coding strategy is the [[Neural_coding#Spike-count_rate|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 is 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:

<math>P(r|s) = \prod_{} P(n_{ij} | s) </math>

where <math>r = n =</math> the number of spikes, <math>n_{ij}</math> is the number of spikes of neuron <math>i</math> at stimulus presentation time <math>j</math>, and s is the stimulus.

=== Instantaneous rate code ===
Adding a small temporal component results in the [[Neural_coding#Time-dependent_firing_rate|spike timing coding]] strategy. Here, the main quantity measured is the number of spikes that occur within a predefined [[Window function|window]] of time T. This method adds another dimension to the previous. This timing code is given by:

<math>P(r|s) = \prod_{l} \left [ \prod_{i,j} v_i ( t_{ijl} | s)dt \right ] exp \left [ -\sum_{i} \int_{0}^{T} dtv_i(t | s) \right ] </math>

where <math>t_{ijl}</math> is the jth spike on the lth presentation of neuron i, <math>v_i(t|s)</math> 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 ===
[[Neural_coding#Temporal_coding|Temporal correlation code]], as the name states, adds [[correlations]] between individual spikes. This means that the time between a spike <math>t_i</math> and its preceding spike <math>t_{i-1}</math> is included. This is given by:

<math>P(r|s) = \prod_{l} \left [ \prod_{i,j} v_i ( t_{ijl},\tau(t_{ijl}) | s)dt \right ] exp \left [ -\sum_{i} \int_{0}^{T} dtv_i(t, \tau(t) | s) \right ] </math>

where <math>\tau(t)</math> is the time interval between a neurons spike and the one preceding it.

=== Ising decoder ===
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,<ref name="Ising">{{cite journal |author=Schaub MT, Schultz SR |title=The Ising decoder: reading out the activity of large neural ensembles |journal=J Comput Neurosci |volume=32 |issue=1 |pages=101–18 |year=2012 |month=February |pmid=21667155 |doi=10.1007/s10827-011-0342-z }}</ref> and is given by:

<math>P(r|s) = \frac{1}{\Zeta(s)} exp \left ( \sum_{i} h_i(s)r_i + \frac{1}{2} \sum_{i\ne j} J_{ij}(s)r_ir_j \right )</math>

where <math>r = (r_1, r_2, . . . , r_n )^T</math> is the set of binary responses of neuron i, <math>h_i</math> is the [[Mean field theory|external fields function]], <math>J_{ij}</math> is the [[Ising_model#Pairwise_correlated_bits|pairwise couplings function]], and <math>\Zeta(s)</math> is the [[partition function]]{{disambiguation needed|date=February 2013}}.

== Agent-based decoding ==
In addition to the probabilistic approach, [[agent-based model]]s 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 [[Chemical synapse|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.<ref name="Hawkins">Hawkins, J., Ahmad, S., & Dubinsky, D. (2006). Hierarchical temporal memory: Concepts, theory and terminology. Whitepaper.</ref><ref name="Hawkins2">Hawkins, J., & Blakeslee, S. (2005). On intelligence. Owl Books.</ref>

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 ==
The advancement in our understanding of neural decoding benefits the development of [[brain-machine interfaces]], [[prosthetics]]<ref name="Donoghue">{{cite journal |author=Donoghue JP |title=Connecting cortex to machines: recent advances in brain interfaces |journal=Nat. Neurosci. |volume=5 |issue=Suppl |pages=1085–8 |year=2002 |month=November |pmid=12403992 |doi=10.1038/nn947 }}</ref> and the understanding of neurological disorders such as [[epilepsy]].<ref name="Gross">{{cite journal |author=Rolston JD, Desai SA, Laxpati NG, Gross RE |title=Electrical stimulation for epilepsy: experimental approaches |journal=Neurosurg. Clin. N. Am. |volume=22 |issue=4 |pages=425–42, v |year=2011 |month=October |pmid=21939841 |pmc=3190668 |doi=10.1016/j.nec.2011.07.010 |url=http://linkinghub.elsevier.com/retrieve/pii/S1042-3680(11)00076-3}}</ref>

== References ==
{{reflist}}

== See also ==
*[[Rate coding]]
*[[Sparse coding]]
*[[Phase-of-firing code]]
*[[Population coding]]
*[[Temporal coding]]
*[[Correlation coding]]
*[[Independent-spike coding]]
*[[NeuroElectroDynamics]]
*[[Neural synchronization]]
*[[Multielectrode array]]
*[[Patch clamp]]
*[[Grandmother Cell]]
*[[Nervous system network models]]
*[[Bursting]]

[[Category:Neural coding| ]]
[[Category:Computational neuroscience]]
[[Category:Neuroscience]]
[[Category:Neural networks]]

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Hierarchical matrix