Complex cell: Difference between revisions
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Inserted Discovery, Simple vs. Complex Cells and Receptive Fields, and Modeling Complex Cells section |
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{{Short description|Complex}} |
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'''Complex cells''' can be found in the [[visual cortex|primary visual cortex (V1)]],<ref name="Hubel1962">{{cite journal| |
'''Complex cells''' can be found in the [[visual cortex|primary visual cortex (V1)]],<ref name="Hubel1962">{{cite journal | vauthors = Hubel DH, Wiesel TN | title = Receptive fields, binocular interaction and functional architecture in the cat's visual cortex | journal = The Journal of Physiology | volume = 160 | pages = 106–54 | date = January 1962 | issue = 1 | pmid = 14449617 | pmc = 1359523 | doi = 10.1113/jphysiol.1962.sp006837 | url = http://www.jphysiol.org/cgi/pmidlookup?view=long&pmid=14449617}}</ref> the [[visual cortex#V2|secondary visual cortex (V2)]], and [[Brodmann area 19]] ([[visual cortex#Third visual cortex, including area V3|V3]]).<ref>{{cite journal | vauthors = Hubel DH, Wiesel TN | title = Receptive Fields and Functional Architecture in Two Nonstriate Visual Areas (18 and 19) of the Cat | journal = Journal of Neurophysiology | volume = 28 | pages = 229–89 | date = March 1965 | issue = 2 | pmid = 14283058 | doi = 10.1152/jn.1965.28.2.229 | url = http://jn.physiology.org/cgi/pmidlookup?view=long&pmid=14283058 }}</ref> |
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Like a [[simple cell]], a complex cell will respond primarily to oriented edges and gratings, however it has a degree of [[translational invariance|spatial invariance]]. This means that its [[receptive field]] cannot be mapped into fixed excitatory and inhibitory zones. Rather, it will respond to patterns of light in a certain orientation within a large receptive field, regardless of the exact location. Some complex cells respond optimally only to movement in a certain direction. |
Like a [[simple cell]], a complex cell will respond primarily to oriented edges and gratings, however it has a degree of [[translational invariance|spatial invariance]]. This means that its [[receptive field]] cannot be mapped into fixed excitatory and inhibitory zones. Rather, it will respond to patterns of light in a certain orientation within a large receptive field, regardless of the exact location. Some complex cells respond optimally only to movement in a certain direction. |
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These cells were discovered by [[Torsten Wiesel]] and [[David Hubel]] in the early 1960s.<ref name="Hubel1962" /> They refrained from reporting on the complex cells in (Hubel 1959) because they did not feel that they understood them well enough at the time.<ref>{{cite book| |
These cells were discovered by [[Torsten Wiesel]] and [[David Hubel]] in the early 1960s.<ref name="Hubel1962" /> They refrained from reporting on the complex cells in (Hubel 1959) because they did not feel that they understood them well enough at the time.<ref>{{cite book|last1=Wiesel|first1=David H. |last2=Hubel |first2=Torsten N.|title=Brain and visual perception : the story of a 25-year collaboration|date=2005|publisher=Oxford University Press|location=New York, N.Y.|isbn=978-0-19-517618-6|edition=[Online-Ausg.].}}</ref><!--Original citation: |
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<ref>''Brain and Visual Perception: The Story of 25-Year Collaboration'' D. H. Hubel and T. N. Wiesel Oxford 2005</ref> |
<ref>''Brain and Visual Perception: The Story of 25-Year Collaboration'' D. H. Hubel and T. N. Wiesel Oxford 2005</ref> |
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--> In Hubel and Wiesel (1962),<ref name="Hubel1962" /> they reported that complex cells were intermixed with simple cells and when excitatory and inhibitory regions could be established, the summation and mutual antagonism properties didn't hold. |
--> In Hubel and Wiesel (1962),<ref name="Hubel1962" /> they reported that complex cells were intermixed with simple cells and when excitatory and inhibitory regions could be established, the summation and mutual antagonism properties didn't hold. |
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The difference between the receptive fields and the characteristics of simple and complex cells is the hierarchical convergent nature of visual processing. Complex cells receive inputs from a number of simple cells. Their receptive field is therefore a summation and integration of the receptive fields of many input simple cells, although some input is directly received from the [[LGN]].<ref>{{cite book |
The difference between the receptive fields and the characteristics of simple and complex cells is the hierarchical convergent nature of visual processing. Complex cells receive inputs from a number of simple cells. Their receptive field is therefore a summation and integration of the receptive fields of many input simple cells, although some input is directly received from the [[LGN]].<ref>{{cite book | last1 = Palmer | first1 = Stephen E. | title = Vision Science: Photons to Phenomenology | location = Cambridge | publisher = [[The MIT Press]] | page = 153 | date = 1999 | isbn = 978-0-262-16183-1 }}</ref> The manner through which simple cells are able to make up complex cells is not fully understood. A simple addition of receptive fields would result in complex cells manifesting observable, separate excitatory/inhibitory regions, which is not the case. |
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| last1 = Palmer | first1 = Stephen E. |
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| title = Vision Science: Photons to Phenomenology |
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| location = Cambridge |
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| publisher = [[The MIT Press]] |
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| page = 153 |
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| date = 1999 |
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| isbn = 978-0-262-16183-1 }}</ref> |
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== Discovery == |
== Discovery == |
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The discovery of the complex cells in visual cortex began with experiments on a cat. Kuffler first |
The discovery of the complex cells in visual cortex began with experiments on a cat. [[Stephen Kuffler|Kuffler]] first shone small spots of light on a cat's retina.<ref name=":2" /> With this, he was able to conclude that ganglion cells have concentric (active at high light levels) receptive fields. These cells also have either an on-center receptive field (excited when the stimulus is presented directly on the center of the receptive field) or off-center receptive field (excited when the stimulus is presented off the center of the receptive field).<ref name=":2">{{cite journal | vauthors = Hubel DH, Wiesel TN | title = Receptive fields of single neurones in the cat's striate cortex | journal = The Journal of Physiology | volume = 148 | issue = 3 | pages = 574–91 | date = October 1959 | pmid = 14403679 | pmc = 1363130 | doi=10.1113/jphysiol.1959.sp006308}}</ref> Later, Hubel and Wiesel began their own experiments on cats to further solidify the knowledge on the visual receptive fields. One experiment recorded from anesthetized cats; these cats were paralyzed to stabilize their eyes. The cat then faced a screen where various patterns of white light were shone. Each cell's receptive fields were mapped for both eyes on sheets of paper.<ref name=":0">{{cite journal | vauthors = Hubel DH, Wiesel TN | title = Receptive fields, binocular interaction and functional architecture in the cat's visual cortex | journal = The Journal of Physiology | volume = 160 | issue = 1 | pages = 106–54 | date = January 1962 | pmid = 14449617 | pmc = 1359523 | doi=10.1113/jphysiol.1962.sp006837}}</ref> |
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Other studies of complex cells have been performed by Movshon et al.,<ref name=movshon/> Emerson et al.,<ref name="emerson">{{cite journal | vauthors = Emerson RC, Citron MC, Vaughn WJ, Klein SA | title = Nonlinear directionally selective subunits in complex cells of cat striate cortex | journal = Journal of Neurophysiology | volume = 58 | pages = 33–65 | date = 1987 | issue = 1 | doi = 10.1152/jn.1987.58.1.33| pmid = 3039079 }}</ref> Touryan et al.<ref name="touryan02">{{cite journal | vauthors = Touryan J, Lau B, Dan Y | title = Isolation of relevant visual features from random stimuli for cortical complex cells | journal = Journal of Neuroscience | volume = 22 | pages = 10811–10818 | date = 2002 | issue = 24 | doi = 10.1523/jneurosci.22-24-10811.2002| pmid = 12486174 | pmc = 6758424 | doi-access = free }}</ref><ref name="touryan05">{{cite journal | vauthors = Touryan J, Felsen G, Dan Y | title = Spatial structure of complex cell receptive fields measured with natural images | journal = Neuron | volume = 45 | pages = 781–791 | date = 2005 | issue = 5 | doi = 10.1016/j.neuron.2005.01.029| pmid = 15748852 | doi-access = free }}</ref> and Rust et al.<ref name="rust05">{{cite journal | vauthors = Rust NC, Schwartz O, Movshon JA, Simoncelli EP | title = Spatiotemporal elements of macaque V1 receptive fields | journal = Neuron | volume = 46 | pages = 945–956 | date = 2005 | issue = 6 | doi = 10.1016/j.neuron.2005.05.021| pmid = 15953422 | doi-access = free }}</ref> |
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== Simple vs. Complex Cells and Receptive Fields == |
== Simple vs. Complex Cells and Receptive Fields == |
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With simple cells and simple receptive fields, the cells in visual cortex could respond in a way that can be noted from arrangements of excitatory and inhibitory regions in their receptive fields. What this means, essentially, is that the receptive fields are |
With simple cells and simple receptive fields, the cells in visual cortex could respond in a way that can be noted from arrangements of excitatory and inhibitory regions in their receptive fields. What this means, essentially, is that the receptive fields are "simple" because there appears to be a relationship between the response of the cell and the receptive field mapped with small spots. Complex cells and complex receptive fields, on the other hand have a more complex response that does not exhibit that relationship. The results from the above experiment determined that simple fields have clear excitatory and inhibitory divisions, where light shone on an excitatory region increases the firing of a cell and light shone on an inhibitory region decreased firing of a cell. There is also evidence of summation properties, such as light shone across a larger region of either division resulted in a greater change in firing rate than light shone across a smaller region. It is also important to note that excitatory regions can inhibit inhibitory regions and vice versa, as well as it is possible to predict responses of the cells from a map of these areas. |
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On the contrary, complex cells and complex receptive fields are defined to be |
On the contrary, complex cells and complex receptive fields are defined to be "not simple." These cell's response to a stimulus cannot be predicted as simple cells can, as they have no inhibitory and excitatory areas. Summation and the inhibition idea also do not often hold. For example, a horizontal slit was presented in the experiment, and it was found that a cell responded highly to this slit. On these complex cells, as long as the slit was horizontal, it did not matter where the slit was positioned on the receptive field. With simple cells, it would be expected that there would be a higher response to a wide slit. However, the opposite effect occurred: the firing of the cell actually decreased. It was also tested for orientation of the slit. For simple cells, it would be expected that as long as the slit covers the excitatory field, the orientation should not matter. Again, the opposite occurred where even slight tilts to the slit resulted in decreased response.<ref name=":0" /> |
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== Modeling Complex Cells == |
== Modeling Complex Cells == |
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From various studies, including Movshon ''et al.'' in 1978 and even as early as 1960s, simple cells have been able to be modeled with a linear model. This would indicate that these simple cells undergo processes that calculate weighted sums of stimulus intensities where the weights are found from the receptive field. This stems from research by |
From various studies, including [[J. Anthony Movshon|Movshon]] ''et al.'' in 1978 and even as early as the 1960s, simple cells have been able to be modeled with a linear model.<ref name=movshon>{{cite journal | vauthors = Movshon JA, Thompson ID, Tolhurst DJ | title = Receptive field organization of complex cells in the cat's striate cortex | journal = The Journal of Physiology | volume = 283 | pages = 79–99 | date = October 1978 | pmid = 722592 | pmc = 1282766 | doi=10.1113/jphysiol.1978.sp012489}}</ref> This would indicate that these simple cells undergo processes that calculate weighted sums of stimulus intensities where the weights are found from the receptive field. This stems from research by [[Christina Enroth-Cugell|Enroth-Cugell]] & Robson in 1966 which modeled ganglion cells similar to P cells in primates (X cells) and ganglion cells similar to M cells in primates (Y cells).<ref>{{cite journal | vauthors = Enroth-Cugell C, Robson JG | title = The contrast sensitivity of retinal ganglion cells of the cat | journal = The Journal of Physiology | volume = 187 | issue = 3 | pages = 517–52 | date = December 1966 | pmid = 16783910 | pmc = 1395960 | doi=10.1113/jphysiol.1966.sp008107}}</ref> Complex cells, on the other hand, are more complex and fall under a different model. Rather, it was noted that these cells perform nonlinear operations, which suggested that they have linear receptive fields, but instead sum a distorted output of subunits. It was found that complex cells shared similarities to Y cells, making this subunit model a promising candidate to model complex cells. |
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Movshon ''et al.'' in 1978 tested responses from simple cells to determine if the simple model for the X cells was a good fit. They later applied the same testing to complex cells, but used the Y cell (subunit) model instead. This model stated that each subunits could respond differently, but the converted responses would be offset in time, so it would sum to a constant value. It also stated that the response of the cells could not be predicted from the receptive field on its own. |
Movshon ''et al.'' in 1978 tested responses from simple cells to determine if the simple model for the X cells was a good fit. They later applied the same testing to complex cells, but used the Y cell (subunit) model instead. This model stated that each subunits could respond differently, but the converted responses would be offset in time, so it would sum to a constant value. It also stated that the response of the cells could not be predicted from the receptive field on its own. Complex cells appeared to match the subunit model, but still lacked the restriction that the receptive fields are linear. This was also tested by measuring the response of a cell when the stimulus contains two bars, which would help show the properties of the receptive field subunit. What they found was that by knowing these properties of the subunits, it was possible to predict spatial frequency selectivity, as was the case for simple cells.<ref name=carandini>{{cite journal | vauthors = Carandini M | title = What simple and complex cells compute | journal = The Journal of Physiology | volume = 577 | issue = Pt 2 | pages = 463–6 | date = December 2006 | pmid = 16973710 | pmc = 1890437 | doi = 10.1113/jphysiol.2006.118976 }}</ref> Hence, complex cells could be modeled by the subunit model used for Y ganglion cells. |
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Other computational models of complex cells have been proposed by Adelson and Bergen,<ref name=adelson>{{cite journal | vauthors = Adelson E, Bergen J | title = Spatiotemporal energy models for the perception of motion | journal = Journal of the Optical Society of America A | volume = 2 | pages = 284–299 | date = 1985 | issue = 2 | doi = 10.1364/josaa.2.000284| pmid = 3973762 | bibcode = 1985JOSAA...2..284A }}</ref> Heeger,<ref name=heeger>{{cite journal | vauthors = Heeger DJ | title = Normalization of cell responses in cat striate cortex | journal = Vis. Neurosci. | volume = 9 | pages = 181–197 | date = 1992 | issue = 2 | doi = 10.1017/s0952523800009640| pmid = 1504027 | s2cid = 22804285 }}</ref> Serre and Riesenhuber,<ref name=serre>{{cite journal | vauthors = Serre T, Riesenhuber M | title = Realistic modeling of simple and complex cell tuning in the HMAX model, and implications for invariant object recognition in cortex | journal = Technical Report AI Memo 2004-017, MIT Computer Science and Artificial Intelligence Laboratory | date = 2004}}</ref> Einhäuser et al.,<ref name=einhauser>{{cite journal | vauthors = Einhäuser W, Kayser C, König P, Körding KP | title = Learning the invariance properties of complex cells from their responses to natural stimuli | journal = European Journal of Neuroscience | volume = 15 | pages = 475–486 | date = 2004| issue = 3 | doi = 10.1046/j.0953-816x.2001.01885.x | pmid = 11876775 | s2cid = 7235110 | doi-access = free }}</ref> Kording et al.,<ref name=kording>{{cite journal | vauthors = Kording KP, Kayser C, Einhäuser W, Konig P | title = How are complex cell properties adapted to the statistics of natural stimuli? | journal = Journal of Neurophysiology | volume = 91 | pages = 206–212 | date = 2004 | issue = 1 | doi = 10.1152/jn.00149.2003 | pmid = 12904330 | hdl = 11858/00-001M-0000-0013-DA31-6 | hdl-access = free }}</ref> Merolla and Boahen,<ref name=merolla>{{cite journal | vauthors = Merolla P, Boahn K | title = A recurrent model of orientation maps with simple and complex cells | journal = Advances in Neural Information Processing Systems (NIPS 2004) | pages = 995–1002 | date = 2004}}</ref> Berkes and Wiscott,<ref name=berkes>{{cite journal | vauthors = Berkes P, Wiskott L | title = Slow feature analysis yields a rich repertoire of complex cell properties | journal = Journal of Vision | volume = 5 | pages = 579–602 | date = 2005| issue = 6 | doi = 10.1167/5.6.9 | pmid = 16097870 | doi-access = free }}</ref> Carandini,<ref name=carandini/> Hansard and Horaud<ref name=hansard>{{cite journal | vauthors = Hansard M, Horaud R | title = A differential model of the complex cell | journal = Neural Computation| volume = 23 | pages = 2324–2357 | date = 2011 | issue = 9 | doi = 10.1162/neco_a_00163| pmid = 21671791 | arxiv = 2012.09027 | s2cid = 2075062 }}</ref> and Lindeberg.<ref name=lindeberg>{{cite journal | vauthors = Lindeberg T | title = Provably scale-covariant continuous hierarchical networks based on scale-normalized differential expressions coupled in cascade | journal = Journal of Mathematical Imaging and Vision | volume = 62 | pages = 120–128 | date = 2020 | doi = 10.1007/s10851-019-00915-x | arxiv=1905.13555| doi-access = free }}</ref> |
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== References == |
== References == |
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{{reflist}} |
{{reflist}} |
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{{DEFAULTSORT:Complex cell}} |
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[[Category:Cerebrum]] |
[[Category:Cerebrum]] |
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[[Category:Visual system]] |
[[Category:Visual system]] |
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{{neuroanatomy-stub}} |
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Latest revision as of 01:16, 11 January 2024
Complex cells can be found in the primary visual cortex (V1),[1] the secondary visual cortex (V2), and Brodmann area 19 (V3).[2]
Like a simple cell, a complex cell will respond primarily to oriented edges and gratings, however it has a degree of spatial invariance. This means that its receptive field cannot be mapped into fixed excitatory and inhibitory zones. Rather, it will respond to patterns of light in a certain orientation within a large receptive field, regardless of the exact location. Some complex cells respond optimally only to movement in a certain direction.
These cells were discovered by Torsten Wiesel and David Hubel in the early 1960s.[1] They refrained from reporting on the complex cells in (Hubel 1959) because they did not feel that they understood them well enough at the time.[3] In Hubel and Wiesel (1962),[1] they reported that complex cells were intermixed with simple cells and when excitatory and inhibitory regions could be established, the summation and mutual antagonism properties didn't hold.
The difference between the receptive fields and the characteristics of simple and complex cells is the hierarchical convergent nature of visual processing. Complex cells receive inputs from a number of simple cells. Their receptive field is therefore a summation and integration of the receptive fields of many input simple cells, although some input is directly received from the LGN.[4] The manner through which simple cells are able to make up complex cells is not fully understood. A simple addition of receptive fields would result in complex cells manifesting observable, separate excitatory/inhibitory regions, which is not the case.
Discovery
[edit]The discovery of the complex cells in visual cortex began with experiments on a cat. Kuffler first shone small spots of light on a cat's retina.[5] With this, he was able to conclude that ganglion cells have concentric (active at high light levels) receptive fields. These cells also have either an on-center receptive field (excited when the stimulus is presented directly on the center of the receptive field) or off-center receptive field (excited when the stimulus is presented off the center of the receptive field).[5] Later, Hubel and Wiesel began their own experiments on cats to further solidify the knowledge on the visual receptive fields. One experiment recorded from anesthetized cats; these cats were paralyzed to stabilize their eyes. The cat then faced a screen where various patterns of white light were shone. Each cell's receptive fields were mapped for both eyes on sheets of paper.[6]
Other studies of complex cells have been performed by Movshon et al.,[7] Emerson et al.,[8] Touryan et al.[9][10] and Rust et al.[11]
Simple vs. Complex Cells and Receptive Fields
[edit]With simple cells and simple receptive fields, the cells in visual cortex could respond in a way that can be noted from arrangements of excitatory and inhibitory regions in their receptive fields. What this means, essentially, is that the receptive fields are "simple" because there appears to be a relationship between the response of the cell and the receptive field mapped with small spots. Complex cells and complex receptive fields, on the other hand have a more complex response that does not exhibit that relationship. The results from the above experiment determined that simple fields have clear excitatory and inhibitory divisions, where light shone on an excitatory region increases the firing of a cell and light shone on an inhibitory region decreased firing of a cell. There is also evidence of summation properties, such as light shone across a larger region of either division resulted in a greater change in firing rate than light shone across a smaller region. It is also important to note that excitatory regions can inhibit inhibitory regions and vice versa, as well as it is possible to predict responses of the cells from a map of these areas.
On the contrary, complex cells and complex receptive fields are defined to be "not simple." These cell's response to a stimulus cannot be predicted as simple cells can, as they have no inhibitory and excitatory areas. Summation and the inhibition idea also do not often hold. For example, a horizontal slit was presented in the experiment, and it was found that a cell responded highly to this slit. On these complex cells, as long as the slit was horizontal, it did not matter where the slit was positioned on the receptive field. With simple cells, it would be expected that there would be a higher response to a wide slit. However, the opposite effect occurred: the firing of the cell actually decreased. It was also tested for orientation of the slit. For simple cells, it would be expected that as long as the slit covers the excitatory field, the orientation should not matter. Again, the opposite occurred where even slight tilts to the slit resulted in decreased response.[6]
Modeling Complex Cells
[edit]From various studies, including Movshon et al. in 1978 and even as early as the 1960s, simple cells have been able to be modeled with a linear model.[7] This would indicate that these simple cells undergo processes that calculate weighted sums of stimulus intensities where the weights are found from the receptive field. This stems from research by Enroth-Cugell & Robson in 1966 which modeled ganglion cells similar to P cells in primates (X cells) and ganglion cells similar to M cells in primates (Y cells).[12] Complex cells, on the other hand, are more complex and fall under a different model. Rather, it was noted that these cells perform nonlinear operations, which suggested that they have linear receptive fields, but instead sum a distorted output of subunits. It was found that complex cells shared similarities to Y cells, making this subunit model a promising candidate to model complex cells.
Movshon et al. in 1978 tested responses from simple cells to determine if the simple model for the X cells was a good fit. They later applied the same testing to complex cells, but used the Y cell (subunit) model instead. This model stated that each subunits could respond differently, but the converted responses would be offset in time, so it would sum to a constant value. It also stated that the response of the cells could not be predicted from the receptive field on its own. Complex cells appeared to match the subunit model, but still lacked the restriction that the receptive fields are linear. This was also tested by measuring the response of a cell when the stimulus contains two bars, which would help show the properties of the receptive field subunit. What they found was that by knowing these properties of the subunits, it was possible to predict spatial frequency selectivity, as was the case for simple cells.[13] Hence, complex cells could be modeled by the subunit model used for Y ganglion cells.
Other computational models of complex cells have been proposed by Adelson and Bergen,[14] Heeger,[15] Serre and Riesenhuber,[16] Einhäuser et al.,[17] Kording et al.,[18] Merolla and Boahen,[19] Berkes and Wiscott,[20] Carandini,[13] Hansard and Horaud[21] and Lindeberg.[22]
References
[edit]- ^ a b c Hubel DH, Wiesel TN (January 1962). "Receptive fields, binocular interaction and functional architecture in the cat's visual cortex". The Journal of Physiology. 160 (1): 106–54. doi:10.1113/jphysiol.1962.sp006837. PMC 1359523. PMID 14449617.
- ^ Hubel DH, Wiesel TN (March 1965). "Receptive Fields and Functional Architecture in Two Nonstriate Visual Areas (18 and 19) of the Cat". Journal of Neurophysiology. 28 (2): 229–89. doi:10.1152/jn.1965.28.2.229. PMID 14283058.
- ^ Wiesel, David H.; Hubel, Torsten N. (2005). Brain and visual perception : the story of a 25-year collaboration ([Online-Ausg.]. ed.). New York, N.Y.: Oxford University Press. ISBN 978-0-19-517618-6.
- ^ Palmer, Stephen E. (1999). Vision Science: Photons to Phenomenology. Cambridge: The MIT Press. p. 153. ISBN 978-0-262-16183-1.
- ^ a b Hubel DH, Wiesel TN (October 1959). "Receptive fields of single neurones in the cat's striate cortex". The Journal of Physiology. 148 (3): 574–91. doi:10.1113/jphysiol.1959.sp006308. PMC 1363130. PMID 14403679.
- ^ a b Hubel DH, Wiesel TN (January 1962). "Receptive fields, binocular interaction and functional architecture in the cat's visual cortex". The Journal of Physiology. 160 (1): 106–54. doi:10.1113/jphysiol.1962.sp006837. PMC 1359523. PMID 14449617.
- ^ a b Movshon JA, Thompson ID, Tolhurst DJ (October 1978). "Receptive field organization of complex cells in the cat's striate cortex". The Journal of Physiology. 283: 79–99. doi:10.1113/jphysiol.1978.sp012489. PMC 1282766. PMID 722592.
- ^ Emerson RC, Citron MC, Vaughn WJ, Klein SA (1987). "Nonlinear directionally selective subunits in complex cells of cat striate cortex". Journal of Neurophysiology. 58 (1): 33–65. doi:10.1152/jn.1987.58.1.33. PMID 3039079.
- ^ Touryan J, Lau B, Dan Y (2002). "Isolation of relevant visual features from random stimuli for cortical complex cells". Journal of Neuroscience. 22 (24): 10811–10818. doi:10.1523/jneurosci.22-24-10811.2002. PMC 6758424. PMID 12486174.
- ^ Touryan J, Felsen G, Dan Y (2005). "Spatial structure of complex cell receptive fields measured with natural images". Neuron. 45 (5): 781–791. doi:10.1016/j.neuron.2005.01.029. PMID 15748852.
- ^ Rust NC, Schwartz O, Movshon JA, Simoncelli EP (2005). "Spatiotemporal elements of macaque V1 receptive fields". Neuron. 46 (6): 945–956. doi:10.1016/j.neuron.2005.05.021. PMID 15953422.
- ^ Enroth-Cugell C, Robson JG (December 1966). "The contrast sensitivity of retinal ganglion cells of the cat". The Journal of Physiology. 187 (3): 517–52. doi:10.1113/jphysiol.1966.sp008107. PMC 1395960. PMID 16783910.
- ^ a b Carandini M (December 2006). "What simple and complex cells compute". The Journal of Physiology. 577 (Pt 2): 463–6. doi:10.1113/jphysiol.2006.118976. PMC 1890437. PMID 16973710.
- ^ Adelson E, Bergen J (1985). "Spatiotemporal energy models for the perception of motion". Journal of the Optical Society of America A. 2 (2): 284–299. Bibcode:1985JOSAA...2..284A. doi:10.1364/josaa.2.000284. PMID 3973762.
- ^ Heeger DJ (1992). "Normalization of cell responses in cat striate cortex". Vis. Neurosci. 9 (2): 181–197. doi:10.1017/s0952523800009640. PMID 1504027. S2CID 22804285.
- ^ Serre T, Riesenhuber M (2004). "Realistic modeling of simple and complex cell tuning in the HMAX model, and implications for invariant object recognition in cortex". Technical Report AI Memo 2004-017, MIT Computer Science and Artificial Intelligence Laboratory.
- ^ Einhäuser W, Kayser C, König P, Körding KP (2004). "Learning the invariance properties of complex cells from their responses to natural stimuli". European Journal of Neuroscience. 15 (3): 475–486. doi:10.1046/j.0953-816x.2001.01885.x. PMID 11876775. S2CID 7235110.
- ^ Kording KP, Kayser C, Einhäuser W, Konig P (2004). "How are complex cell properties adapted to the statistics of natural stimuli?". Journal of Neurophysiology. 91 (1): 206–212. doi:10.1152/jn.00149.2003. hdl:11858/00-001M-0000-0013-DA31-6. PMID 12904330.
- ^ Merolla P, Boahn K (2004). "A recurrent model of orientation maps with simple and complex cells". Advances in Neural Information Processing Systems (NIPS 2004): 995–1002.
- ^ Berkes P, Wiskott L (2005). "Slow feature analysis yields a rich repertoire of complex cell properties". Journal of Vision. 5 (6): 579–602. doi:10.1167/5.6.9. PMID 16097870.
- ^ Hansard M, Horaud R (2011). "A differential model of the complex cell". Neural Computation. 23 (9): 2324–2357. arXiv:2012.09027. doi:10.1162/neco_a_00163. PMID 21671791. S2CID 2075062.
- ^ Lindeberg T (2020). "Provably scale-covariant continuous hierarchical networks based on scale-normalized differential expressions coupled in cascade". Journal of Mathematical Imaging and Vision. 62: 120–128. arXiv:1905.13555. doi:10.1007/s10851-019-00915-x.