2016, 13(3): 579-596. doi: 10.3934/mbe.2016009

On the properties of input-to-output transformations in neuronal networks

1. 

School of Science and Technology, Georgia Gwinnett College, Lawrenceville, GA 30043, United States

2. 

Department of Mathematics and Informatics, Université des Antilles, Pointe-à-Pitre, Guadeloupe, France

Received  March 2015 Revised  October 2015 Published  January 2016

Information processing in neuronal networks in certain important cases can be considered as maps of binary vectors, where ones (spikes) and zeros (no spikes) of input neurons are transformed into spikes and no spikes of output neurons. A simple but fundamental characteristic of such a map is how it transforms distances between input vectors into distances between output vectors. We advanced earlier known results by finding an exact solution to this problem for McCulloch-Pitts neurons. The obtained explicit formulas allow for detailed analysis of how the network connectivity and neuronal excitability affect the transformation of distances in neurons. As an application, we explored a simple model of information processing in the hippocampus, a brain area critically implicated in learning and memory. We found network connectivity and neuronal excitability parameter values that optimize discrimination between similar and distinct inputs. A decrease of neuronal excitability, which in biological neurons may be associated with decreased inhibition, impaired the optimality of discrimination.
Citation: Andrey Olypher, Jean Vaillant. On the properties of input-to-output transformations in neuronal networks. Mathematical Biosciences & Engineering, 2016, 13 (3) : 579-596. doi: 10.3934/mbe.2016009
References:
[1]

P. Andersen, R. Morris, D. Amaral T. Bliss and J. O'Keefe, Historical perspective: Proposed functions, biological characteristics, and neurobiological models of the hippocampus, in The Hippocampus Book (eds. P. Andersen, R. Morris, D. Amaral, T. Bliss and J. O'Keefe), Oxford University Press, 2006, 9-36.

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T. M. Bartol, C. Bromer, J. Kinney, M. A. Chirillo, J. N. Bourne, K. M. Harris and T. J. Sejnowski, Hippocampal spine head sizes are highly precise, preprint, (2015). doi: 10.1101/016329.

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K. W. Bieri, K. N. Bobbitt and L. L. Colgin, Slow and fast gamma rhythms coordinate different spatial coding modes in hippocampal place cells, Neuron, 82 (2014), 670-681. doi: 10.1016/j.neuron.2014.03.013.

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A. Bragin, G. Jando, Z. Nadasdy, J. Hetke, K. Wise and G. Buzsaki, Gamma (40-100 Hz) oscillation in the hippocampus of the behaving rat, J. Neurosci., 15 (1995), 47-60.

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I. H. Brivanlou, J. L. Dantzker, C. F. Stevens and E. M. Callaway, Topographic specificity of functional connections from hippocampal CA3 to CA1, Proc. Natl. Acad. Sci. USA, 101 (2004), 2560-2565. doi: 10.1073/pnas.0308577100.

[6]

N. Brunel, V. Hakim, P. Isope, J. P. Nadal and B. Barbour, Optimal information storage and the distribution of synaptic weights: Perceptron versus Purkinje cell, Neuron, 43 (2004), 745-757. doi: 10.1016/j.neuron.2004.08.023.

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D. Bush, C. Barry and N. Burgess, What do grid cells contribute to place cell firing?, Trends Neurosci., 37 (2014), 136-145. doi: 10.1016/j.tins.2013.12.003.

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S. Cash and R. Yuste, Linear summation of excitatory inputs by CA1 pyramidal neurons, Neuron, 22 (1999), 383-394. doi: 10.1016/S0896-6273(00)81098-3.

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C. K. Chow, On the characterization of threshold functions, Proc. Symposium on Switching Circuit Theory and Logical Design (FOCS), (1961), 34-38. doi: 10.1109/FOCS.1961.24.

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C. Clopath and N. Brunel, Optimal properties of analog perceptrons with excitatory weights, PLoS Comput. Biol., 9 (2013), e1002919. doi: 10.1371/journal.pcbi.1002919.

[11]

L. L. Colgin and E. I. Moser, Gamma oscillations in the hippocampus, Physiology (Bethesda), 25 (2010), 319-329. doi: 10.1152/physiol.00021.2010.

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V. Cutsuridis, S. Cobb and B. P. Graham, Encoding and retrieval in a model of the hippocampal CA1 microcircuit, Hippocampus, 20 (2010), 423-446. doi: 10.1002/hipo.20661.

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W. Feller, An Introduction to Probability Theory and Its Applications, Wiley, New York, NY, 1968.

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A. A. Fenton and R. Muller, Place cell discharge is extremely variable during individual passes of the rat through the firing field, P. Natl. Acad. Sci. USA, 95 (1998), 3182-3187. doi: 10.1073/pnas.95.6.3182.

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M. García-Sanchez and R. Huerta, Design parameters of the fan-out phase of sensory systems, J. Comput. Neurosci., 15 (2003), 5-17.

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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, $2^{nd}$ edition, Addison-Wesley Professional, 1994.

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R. Huerta, Learning pattern recognition and decision making in the insect brain, Proc. of AIP Conf., 1510 (2013), 101-119. doi: 10.1063/1.4776507.

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F. Izsák, Maximum likelihood estimation for constrained parameters of multinomial distributions - Application to Zipf-Mandelbrot models, Comput. Stat. Data Anal., 51 (2006), 1575-1583. doi: 10.1016/j.csda.2006.05.008.

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T. Jarsky, A. Roxin, W. L. Kath and N. Spruston, Conditional dendritic spike propagation following distal synaptic activation of hippocampal CA1 pyramidal neurons, Nat. Neurosci., 8 (2005), 1667-1676. doi: 10.1038/nn1599.

[20]

G. A. Kerchner and R. A. Nicoll, Silent synapses and the emergence of a postsynaptic mechanism for LTP, Nat. Rev. Neurosci., 9 (2008), 813-825.

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R. Kramer, D. Fortin and D. Trauner, New photochemical tools for controlling neuronal activity, Curr. Opin. Neurobiol., 19 (2009), 544-552. doi: 10.1016/j.conb.2009.09.004.

[22]

X. Li and G. A. Ascoli, Effects of synaptic synchrony on the neuronal input-output relationship, Neural. Comput., 20 (2008), 1717-1731. doi: 10.1162/neco.2008.10-06-385.

[23]

C. Loader, Fast and accurate computation of binomial probabilities, 2000. Available from: http://savannah.gnu.org/bugs/download.php?file_id=24016.

[24]

J. C. Magee and E. P. Cook, Somatic epsp amplitude is independent of synapse location in hippocampal pyramidal neurons, Nat. Neurosci., 3 (2000), 895-903.

[25]

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, B. Math. Biophys., 5 (1943), 115-133. doi: 10.1007/BF02478259.

[26]

M. Megias, Z. Emri, T. F. Freund and A. I. Gulyas, Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells, Neuroscience, 102 (2001), 527-540. doi: 10.1016/S0306-4522(00)00496-6.

[27]

E. I. Moser and M. B. Moser, Grid cells and neural coding in high-end cortices, Neuron, 80 (2013), 765-774. doi: 10.1016/j.neuron.2013.09.043.

[28]

A. V. Olypher, P. Lansky and A. A. Fenton, Properties of the extra-positional signal in hippocampal place cell discharge derived from the overdispersion in location-specific firing, Neuroscience, 111 (2002), 553-566. doi: 10.1016/S0306-4522(01)00586-3.

[29]

A. V. Olypher, W. W. Lytton and A. A. Prinz, Transformation of inputs in a model of the rat hippocampal CA1 network, SFN Meeting Planner, 11 (2010), p56. doi: 10.1186/1471-2202-11-S1-P56.

[30]

A. V. Olypher, W. W. Lytton and A. A. Prinz, Input-to-output transformation in a model of the rat hippocampal CA1 network, Front Comput. Neurosci., 6 (2012), p57. doi: 10.3389/fncom.2012.00057.

[31]

A. Olypher and J. Vaillant, On the properties of input-to-output transformations in networks of perceptrons, arXiv:1312.1206, (2013).

[32]

D. Parameshwaran and U. S. Bhalla, Summation in the hippocampal CA3-CA1 network remains robustly linear following inhibitory modulation and plasticity, but undergoes scaling and offset transformations, Front Comput. Neurosci., 6 (2012), p71. doi: 10.3389/fncom.2012.00071.

[33]

J. Perez-Orive, O. Mazor, G. C. Turner, S. Cassenaer, R. I. Wilson and G. Laurent, Oscillations and sparsening of odor representations in the mushroom body, Science, 297 (2002), 359-365. doi: 10.1126/science.1070502.

[34]

M. Smith, G. Ellis-Davies and J. Magee, Mechanism of the distance-dependent scaling of schaffer collateral synapses in rat CA1 pyramidal neurons, J. Physiol., 548 (2003), 245-258.

[35]

T. Solstad, H. N. Yousif and T. J. Sejnowski, Place cell rate remapping by CA3 recurrent collaterals, PLoS Comput. Biol., 10 (2014), e1003648. doi: 10.1371/journal.pcbi.1003648.

[36]

A. Treves and E. T. Rolls, Computational analysis of the role of the hippocampus in memory, Hippocampus, 4 (1994), 374-391. doi: 10.1002/hipo.450040319.

[37]

L. G. Valiant, The hippocampus as a stable memory allocator for cortex, Neural Comput., 24 (2012), 2873-2899. doi: 10.1162/NECO_a_00357.

show all references

References:
[1]

P. Andersen, R. Morris, D. Amaral T. Bliss and J. O'Keefe, Historical perspective: Proposed functions, biological characteristics, and neurobiological models of the hippocampus, in The Hippocampus Book (eds. P. Andersen, R. Morris, D. Amaral, T. Bliss and J. O'Keefe), Oxford University Press, 2006, 9-36.

[2]

T. M. Bartol, C. Bromer, J. Kinney, M. A. Chirillo, J. N. Bourne, K. M. Harris and T. J. Sejnowski, Hippocampal spine head sizes are highly precise, preprint, (2015). doi: 10.1101/016329.

[3]

K. W. Bieri, K. N. Bobbitt and L. L. Colgin, Slow and fast gamma rhythms coordinate different spatial coding modes in hippocampal place cells, Neuron, 82 (2014), 670-681. doi: 10.1016/j.neuron.2014.03.013.

[4]

A. Bragin, G. Jando, Z. Nadasdy, J. Hetke, K. Wise and G. Buzsaki, Gamma (40-100 Hz) oscillation in the hippocampus of the behaving rat, J. Neurosci., 15 (1995), 47-60.

[5]

I. H. Brivanlou, J. L. Dantzker, C. F. Stevens and E. M. Callaway, Topographic specificity of functional connections from hippocampal CA3 to CA1, Proc. Natl. Acad. Sci. USA, 101 (2004), 2560-2565. doi: 10.1073/pnas.0308577100.

[6]

N. Brunel, V. Hakim, P. Isope, J. P. Nadal and B. Barbour, Optimal information storage and the distribution of synaptic weights: Perceptron versus Purkinje cell, Neuron, 43 (2004), 745-757. doi: 10.1016/j.neuron.2004.08.023.

[7]

D. Bush, C. Barry and N. Burgess, What do grid cells contribute to place cell firing?, Trends Neurosci., 37 (2014), 136-145. doi: 10.1016/j.tins.2013.12.003.

[8]

S. Cash and R. Yuste, Linear summation of excitatory inputs by CA1 pyramidal neurons, Neuron, 22 (1999), 383-394. doi: 10.1016/S0896-6273(00)81098-3.

[9]

C. K. Chow, On the characterization of threshold functions, Proc. Symposium on Switching Circuit Theory and Logical Design (FOCS), (1961), 34-38. doi: 10.1109/FOCS.1961.24.

[10]

C. Clopath and N. Brunel, Optimal properties of analog perceptrons with excitatory weights, PLoS Comput. Biol., 9 (2013), e1002919. doi: 10.1371/journal.pcbi.1002919.

[11]

L. L. Colgin and E. I. Moser, Gamma oscillations in the hippocampus, Physiology (Bethesda), 25 (2010), 319-329. doi: 10.1152/physiol.00021.2010.

[12]

V. Cutsuridis, S. Cobb and B. P. Graham, Encoding and retrieval in a model of the hippocampal CA1 microcircuit, Hippocampus, 20 (2010), 423-446. doi: 10.1002/hipo.20661.

[13]

W. Feller, An Introduction to Probability Theory and Its Applications, Wiley, New York, NY, 1968.

[14]

A. A. Fenton and R. Muller, Place cell discharge is extremely variable during individual passes of the rat through the firing field, P. Natl. Acad. Sci. USA, 95 (1998), 3182-3187. doi: 10.1073/pnas.95.6.3182.

[15]

M. García-Sanchez and R. Huerta, Design parameters of the fan-out phase of sensory systems, J. Comput. Neurosci., 15 (2003), 5-17.

[16]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, $2^{nd}$ edition, Addison-Wesley Professional, 1994.

[17]

R. Huerta, Learning pattern recognition and decision making in the insect brain, Proc. of AIP Conf., 1510 (2013), 101-119. doi: 10.1063/1.4776507.

[18]

F. Izsák, Maximum likelihood estimation for constrained parameters of multinomial distributions - Application to Zipf-Mandelbrot models, Comput. Stat. Data Anal., 51 (2006), 1575-1583. doi: 10.1016/j.csda.2006.05.008.

[19]

T. Jarsky, A. Roxin, W. L. Kath and N. Spruston, Conditional dendritic spike propagation following distal synaptic activation of hippocampal CA1 pyramidal neurons, Nat. Neurosci., 8 (2005), 1667-1676. doi: 10.1038/nn1599.

[20]

G. A. Kerchner and R. A. Nicoll, Silent synapses and the emergence of a postsynaptic mechanism for LTP, Nat. Rev. Neurosci., 9 (2008), 813-825.

[21]

R. Kramer, D. Fortin and D. Trauner, New photochemical tools for controlling neuronal activity, Curr. Opin. Neurobiol., 19 (2009), 544-552. doi: 10.1016/j.conb.2009.09.004.

[22]

X. Li and G. A. Ascoli, Effects of synaptic synchrony on the neuronal input-output relationship, Neural. Comput., 20 (2008), 1717-1731. doi: 10.1162/neco.2008.10-06-385.

[23]

C. Loader, Fast and accurate computation of binomial probabilities, 2000. Available from: http://savannah.gnu.org/bugs/download.php?file_id=24016.

[24]

J. C. Magee and E. P. Cook, Somatic epsp amplitude is independent of synapse location in hippocampal pyramidal neurons, Nat. Neurosci., 3 (2000), 895-903.

[25]

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, B. Math. Biophys., 5 (1943), 115-133. doi: 10.1007/BF02478259.

[26]

M. Megias, Z. Emri, T. F. Freund and A. I. Gulyas, Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells, Neuroscience, 102 (2001), 527-540. doi: 10.1016/S0306-4522(00)00496-6.

[27]

E. I. Moser and M. B. Moser, Grid cells and neural coding in high-end cortices, Neuron, 80 (2013), 765-774. doi: 10.1016/j.neuron.2013.09.043.

[28]

A. V. Olypher, P. Lansky and A. A. Fenton, Properties of the extra-positional signal in hippocampal place cell discharge derived from the overdispersion in location-specific firing, Neuroscience, 111 (2002), 553-566. doi: 10.1016/S0306-4522(01)00586-3.

[29]

A. V. Olypher, W. W. Lytton and A. A. Prinz, Transformation of inputs in a model of the rat hippocampal CA1 network, SFN Meeting Planner, 11 (2010), p56. doi: 10.1186/1471-2202-11-S1-P56.

[30]

A. V. Olypher, W. W. Lytton and A. A. Prinz, Input-to-output transformation in a model of the rat hippocampal CA1 network, Front Comput. Neurosci., 6 (2012), p57. doi: 10.3389/fncom.2012.00057.

[31]

A. Olypher and J. Vaillant, On the properties of input-to-output transformations in networks of perceptrons, arXiv:1312.1206, (2013).

[32]

D. Parameshwaran and U. S. Bhalla, Summation in the hippocampal CA3-CA1 network remains robustly linear following inhibitory modulation and plasticity, but undergoes scaling and offset transformations, Front Comput. Neurosci., 6 (2012), p71. doi: 10.3389/fncom.2012.00071.

[33]

J. Perez-Orive, O. Mazor, G. C. Turner, S. Cassenaer, R. I. Wilson and G. Laurent, Oscillations and sparsening of odor representations in the mushroom body, Science, 297 (2002), 359-365. doi: 10.1126/science.1070502.

[34]

M. Smith, G. Ellis-Davies and J. Magee, Mechanism of the distance-dependent scaling of schaffer collateral synapses in rat CA1 pyramidal neurons, J. Physiol., 548 (2003), 245-258.

[35]

T. Solstad, H. N. Yousif and T. J. Sejnowski, Place cell rate remapping by CA3 recurrent collaterals, PLoS Comput. Biol., 10 (2014), e1003648. doi: 10.1371/journal.pcbi.1003648.

[36]

A. Treves and E. T. Rolls, Computational analysis of the role of the hippocampus in memory, Hippocampus, 4 (1994), 374-391. doi: 10.1002/hipo.450040319.

[37]

L. G. Valiant, The hippocampus as a stable memory allocator for cortex, Neural Comput., 24 (2012), 2873-2899. doi: 10.1162/NECO_a_00357.

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