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, (2006), 9. Google Scholar

[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. Google Scholar

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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. doi: 10.1016/j.neuron.2014.03.013. Google Scholar

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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. Google Scholar

[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. doi: 10.1073/pnas.0308577100. Google Scholar

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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. doi: 10.1016/j.neuron.2004.08.023. Google Scholar

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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. doi: 10.1016/j.tins.2013.12.003. Google Scholar

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

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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. doi: 10.1109/FOCS.1961.24. Google Scholar

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

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

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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. doi: 10.1002/hipo.20661. Google Scholar

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W. Feller, An Introduction to Probability Theory and Its Applications,, Wiley, (1968). Google Scholar

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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. doi: 10.1073/pnas.95.6.3182. Google Scholar

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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. Google Scholar

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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science,, $2^{nd}$ edition, (1994). Google Scholar

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

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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. doi: 10.1016/j.csda.2006.05.008. Google Scholar

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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. doi: 10.1038/nn1599. Google Scholar

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

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

[22]

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

[23]

C. Loader, Fast and accurate computation of binomial probabilities,, 2000. Available from: , (). Google Scholar

[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. Google Scholar

[25]

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

[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. doi: 10.1016/S0306-4522(00)00496-6. Google Scholar

[27]

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

[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. doi: 10.1016/S0306-4522(01)00586-3. Google Scholar

[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). doi: 10.1186/1471-2202-11-S1-P56. Google Scholar

[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). doi: 10.3389/fncom.2012.00057. Google Scholar

[31]

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

[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). doi: 10.3389/fncom.2012.00071. Google Scholar

[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. doi: 10.1126/science.1070502. Google Scholar

[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. Google Scholar

[35]

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

[36]

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

[37]

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

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, (2006), 9. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.neuron.2014.03.013. Google Scholar

[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. Google Scholar

[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. doi: 10.1073/pnas.0308577100. Google Scholar

[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. doi: 10.1016/j.neuron.2004.08.023. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

W. Feller, An Introduction to Probability Theory and Its Applications,, Wiley, (1968). Google Scholar

[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. doi: 10.1073/pnas.95.6.3182. Google Scholar

[15]

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

[16]

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science,, $2^{nd}$ edition, (1994). Google Scholar

[17]

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

[18]

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

[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. doi: 10.1038/nn1599. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[23]

C. Loader, Fast and accurate computation of binomial probabilities,, 2000. Available from: , (). Google Scholar

[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. Google Scholar

[25]

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

[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. doi: 10.1016/S0306-4522(00)00496-6. Google Scholar

[27]

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

[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. doi: 10.1016/S0306-4522(01)00586-3. Google Scholar

[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). doi: 10.1186/1471-2202-11-S1-P56. Google Scholar

[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). doi: 10.3389/fncom.2012.00057. Google Scholar

[31]

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

[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). doi: 10.3389/fncom.2012.00071. Google Scholar

[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. doi: 10.1126/science.1070502. Google Scholar

[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. Google Scholar

[35]

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

[36]

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

[37]

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

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