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Do neurons have a voltage or a current threshold for action potential initiation?

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Abstract

The majority of neural network models consider the output of single neurons to be a continuous, positive, and saturating firing ratef(t), while a minority treat neuronal output as a series of delta pulses ∑δ (t — t i ). We here argue that the issue of the proper output representation relates to the biophysics of the cells in question and, in particular, to whether initiation of somatic action potentials occurs when a certain thresholdvoltage or a thresholdcurrent is exceeded. We approach this issue using numerical simulations of the electrical behavior of a layer 5 pyramidal cell from cat visual cortex. The dendritic tree is passive while the cell body includes eight voltage- and calcium-dependent membrane conductances.

We compute both the steady-state (I static (V m )) and the instantaneous (I o (Vm)) I–V relationships and argue that the amplitude of the local maximum inI static (V m ) corresponds to the current thresholdI th for sustained inputs, while the location of the middle zero-crossing ofI o corresponds to a fixed voltage thresholdV th for rapid inputs. We confirm this using numerical simulations: for “rapid” synaptic inputs, spikes are initiated if the somatic potential exceedsV th, while for slowly varying inputI th must be exceeded. Due to the presence of the large dendritic tree, no charge thresholdQ th exists for physiological input.

Introducing the temporal average of the somatic membrane potential 〈(V m)〉 while the cell is spiking repetitively, allows us to define a dynamic I-V relationship dynamic (〈(V m)〉). We find an exponential relationship between 〈(V m)〉 and the net current sunk by the somatic membrane during spiking (diode-like behavior). The slope ofI∞/dynamic(〈(V m))〉 allows us to define a dynamic input conductance and a time constant that characterizes how rapidly the cell changes its output firing frequency in response to a change in its input.

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References

  • Abbott LF (1991) Realistic synaptic inputs for model neuronal networks.Network 2:245–258.

    Google Scholar 

  • Abbott LF and van Wreeswijk C (1993) Asynchronous states in networks of pulse-coupled oscillators.Physical Review E 48(2):1483–1490.

    Google Scholar 

  • Abeles M (1982) Local Cortical Circuits. Springer-Verlag, Berlin.

    Google Scholar 

  • Abeles M, Bergman H, Margalit E, and Vaadia E (1993) Spatiotemporal firing patterns in the frontal-cortex of behaving monkeys.J. Neurophysiol. 70(4): 1629–1638.

    Google Scholar 

  • Agin D (1964) Hodgkin-huxley equations: logarithmic relation between membrane current and frequency of repetitive activity.Nature, Lond. 201:625–626.

    Google Scholar 

  • Agmon A and Connors BW (1992) Correlation between intrinsic firing patterns and thalamocortical synaptic responses of neurons in mouse barrel cortex.J. Neurosci. 12(1):319–329.

    Google Scholar 

  • Ahmed B, Anderson JC, Douglas RJ, Martin KAC, and Whitteridge D (1995) The current-discharge patterns of identified neurons in cat visual cortex.J. Neurophysiology, Submitted.

  • Bernander Ö (1993) Synaptic integration and its control in neocortical pyramidal cells. Ph.D. thesis, California Insitute of Technology.

  • Bernander Ö, Douglas RJ, and Koch C (1994a) Amplifying and linearizing apical synaptic inputs to cortical pyramidal cells. In RP Lippman, JE Moody and DS Touretzky, editors,Neural information processing systems 6. 2929 Campus Drive #260, San Mateo, CA 94403, Morgan Kaufmann, pp. 519–526.

    Google Scholar 

  • Bernander Ö, Douglas RJ, and Koch C (1994b) Amplification and linearization of distal synaptic input to neocortical pyramidal cells.J. Neumphysiology 72:2743–2753.

    Google Scholar 

  • Bernander Ö, Douglas RJ, Martin KAC, and Koch C (1991) Synaptic background activity influences spatiotemporal integration in single pyramidal cells.Proc. Natl. Acad. Sci. USA 88:11569–11573.

    Google Scholar 

  • Churchland PS and Sejnowski TJ (1992) The Computational Brain. MIT Press, Cambridge, Massachusetts.

    Google Scholar 

  • Cole KS (1972) Membranes Ions and Impulses. University of California Press, Berkeley and Los Angeles, California.

    Google Scholar 

  • Connor JA and Stevens CF (1971) Inward and delayed outward membrane currents in isolated neural somata under voltage clamp.J. Physiology 213:31–53.

    Google Scholar 

  • Connors BW and Gutnick MJ (1990) Intrinsic firing patterns of diverse neocortical neurons.Trends Neurosci. 13(3):99–104.

    Google Scholar 

  • Douglas RJ, Mahowald M, and Mead C (1995) Neuromorphic analog VLSI.Ann. Rev. Neurosci. 18, in-Press.

  • Douglas RJ and Martin KAC (1990) Neocortex. In GM Shepherd, editors, The synaptic organization of the brain, 2nd edition. Chapter 12, Oxford University Press, Oxford, pp. 389–438.

    Google Scholar 

  • Douglas RJ, Martin KAC, and Whitteridge D (1991) An intracellular analysis of the visual responses of neuronis in cat visual cortex.J. Physiology 440:659–696.

    Google Scholar 

  • Eckhorn R, Bauer R, Jordan W, Brosch M, Kruse W, Munk M, and Reitböck HJ (1988) Coherent oscillations: a mechanism of feature linking in the visual cortex?Biol Cybern. 60:121–130.

    Google Scholar 

  • Freeman WJ (1975) Mass action in the nervous system. Academic Press.

  • Gray CM, König P, Engel AK, and Singer W(1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties.Nature 338:334–337.

    Google Scholar 

  • Gray CM and Singer W (1989) Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex.Proc. Natl. Acad. Sci. USA 86:1698–1702.

    Google Scholar 

  • Hertz J, Krogh A, and Palmer RG (1991) Introduciton to the Theory of Neural Computation. Addison Wesley.

  • Hille B (1992) Ionic Channels of Excitable Membranes. Sinauer Associates, Inc. Publishers, Sunderland, Massachusetts, second edition.

    Google Scholar 

  • Hines M (1989) A program for simulation of nerve equations with branching geometries.Int. J. Biomed. Comput. 24:55–68.

    Google Scholar 

  • Hodgkin AL and Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve.J. Physiology 117:500–544.

    Google Scholar 

  • Hodgkin AL and Rushton WAH (1946) The electrical constants of a crustacean nerve fibre.Proc. Roy. Soc. Lond. B 133:444–479.

    Google Scholar 

  • Holmes WR, Segev I, and Rall W (1992) Interpretation of time constant and electronic length estimates in multicylinder or branched neuronal structures.J. Neumphysiology 68(4): 1401–1420.

    Google Scholar 

  • Holmes WR and Woody CD (1989) Effects of uniform and non-uniform synaptic activation-distributions on the cable properties of modeled cortical pyramidal neurons.Brain Research 505:12–22.

    Google Scholar 

  • Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons.Proc. Natl. Acad. Sci. USA 81:3088–3092.

    Google Scholar 

  • Hopfield JJ (1994) Neurons, dynamics and computation.Physics Today 47(2):40–46.

    Google Scholar 

  • Hopfield JJ and Tank DW (1985) “Neutral” computation of decisions in optimization problems.Biol. Cyber. 52:141–152.

    Google Scholar 

  • Jack JJB, Noble D, and Tsien RW (1975) Electric current flow in excitable cells. Oxford University Press, Oxford.

    Google Scholar 

  • Kawaguchi Y (1993) Groupings of nonpyramidal and pyramidal cells with specific physiological and morphological characteristics in rat frontal cortex.J. Neurophysiol. 69(2):416–431.

    Google Scholar 

  • Knight B (1972) Dynamics of encoding in a population of neurons,J. Gen. Physiol 59:734–766.

    Google Scholar 

  • Koch C, Marroquin J, and Yuille A (1986) Analog “neuronal” networks in early vision.Proc. Natl. Acad. Sci. USA 83:4263–4267.

    Google Scholar 

  • Koch C and Segev I, (Eds) (1989) Methods in Neuronal Modeling. MIT Press, Cambridge, Massachusetts.

    Google Scholar 

  • König P and Schulen TB (1991) Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization.Neural Camp. 3:155–166.

    Google Scholar 

  • Lapicque L (1907) Recherches quantitatifs sur l'excitation electrique des nerfs traitée comme une polarisation.J. Physiol Paris 9:622–635.

    Google Scholar 

  • Lapicque L (1926) L'excitababilité en fonction du temps. Presses Universitaires de France, Paris, France.

    Google Scholar 

  • Larkman AU (1991) Dendritic morphology of pyramidal neurons of the visual cortex of the rat: III. Spine distributions.J. Comp. Neurology 306:332–343.

    Google Scholar 

  • Lytton WW and Sejnowski TJ (1991) Simulations of cortical pyramidal neurons synchronized by inhibitory interneurons.J. Neurophysiology 66(3):1059–1079.

    Google Scholar 

  • Mahowald M and Douglas RJ (1991) A silicon neuron.Nature 354:515–518.

    Google Scholar 

  • Marder E, Abbott LF, and Buchholtz F (1993) Physiological insights from cellular and network models of the stomatogastric nervous system of lobsters and crabs.American Zoologist 33(1): 29–39.

    Google Scholar 

  • Mason A and Larkman AU (1990) Correlations between morphology and electrophysiology of pyramidal neurons in slices of rat visual cortex. II. Electrophysiology.J. Neuroscience 10(5): 1415–1428.

    Google Scholar 

  • McCormick DA (1992) Neurotransmitter actions in the thalamus and cerebral cortex and their role in neuromodulation of thalamocortical activity.Progress in Neurobiology 39:337–388.

    Google Scholar 

  • McCormick DA, Connors BW, Lighthall JW, and Prince DA (1985) Comparative electrophysiology of pyramidal and sparsely spiny stellate neurons of the neocortex.J. Neurophysiology 54(4): 782–806.

    Google Scholar 

  • McCullough WS and Pitts W (1943) A logical calculus of the ideas immanent in nervous activity.Bull. Math. Biophys. 5:115–133.

    Google Scholar 

  • Noble D and Stein RB (1966) The threshold conditions for initiation of action potentials by excitable cells.J. Physiol. 187:129–162.

    Google Scholar 

  • Optican LM and Richmond B (1987) Temporal encoding of two-dimensional patterns by single units in primate inferior temporal cortex. III. Information theoretic analysis,J. Neurophysiol. 57:162–178.

    Google Scholar 

  • Rapp M, Yarom Y, and Segev I (1992) The impact of parallel fiber background activity on the cable properties of cerebellar Purkinje cells.Neural Computation 4:518–533.

    Google Scholar 

  • Rinzel J (1985) Excitation dynamics: insights from simplified membrane models.Fed. Proc. 44:2944–2946.

    Google Scholar 

  • Rinzel J and Ermentrout BB (1989) Analysis of neural excitability and oscillations. In C Koch and I Segev, eds., Methods in Neuronal Modeling. MIT Press, Cambridge, Massachusetts, Chapter 5, pp. 135–169.

    Google Scholar 

  • Schwindt PC, Spain WJ, and Crill WE (1992) Calcium-dependent potassium currents in neurons from cat sensorimotor cortex.J. Neurophysiology 67(1):216–226.

    Google Scholar 

  • Schwindt PC, Spain WJ, Foehring RC, Stafstrom CE, Chubb MC, and Crill WE (1988) Multiple potassium conductances and their functions in neurons from cat sensorimotor cortex in virto.J. Neurophysiology 59(2):424–449.

    Google Scholar 

  • Segev I, Rapp M, Manor Y, and Yarom Y (1992) Analog and digital processing in single nerve cells: Dendritic integration and axonal propagation. In T McKenna, J Davis and SF Zornetzer, eds., Single Neuron Computation. Academic Press, Boston, Massachusetts, Chapter 7, pp. 173–198.

    Google Scholar 

  • Shadlen MN and Newsome WT (1994) Noise, neural codes and cortical organization.Current Opinion in Neurobiology 4:509–579.

    Google Scholar 

  • Shelton DP (1985) Membrane resistivity estimated for the Purkinje neuron by means of a passive computer model.J. Neuroscience 14(1):111–131.

    Google Scholar 

  • Softky W (1994) Sub-millisecond coincidence detection in active dendritic trees.Neuroscience 58(1):13–41.

    Google Scholar 

  • Softky W and Koch C (1993) The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs.J. Neuroscience 13(1):334–350.

    Google Scholar 

  • Somogyi P (1989) Synaptic organization of GABA-ergic neurons and GABA—A receptors in the lateral geniculate nucleus and visual cortex. In DK Lam and CD Gilbert, eds. Neural Mechanisms of Visual Perception. Portfolio, Houston, pp. 35–62.

    Google Scholar 

  • Spain WJ, Schwindt PC, and Crill WE (1987) Anomalous rectification in neurons from cat sensorimotor cortex in vitro.J. Neurophysiology 57(5):1555–1576.

    Google Scholar 

  • Spain WJ, Schwindt PC, and Crill WE (1991) Two transient potassium currents in layer V pyramidal neurons from cat sensorimotor cortex.J. Physiology 434:591–607.

    Google Scholar 

  • Spruston N, Jaffe DB, and Johnston D (1994) Dendritic attenuation of synaptic potentials and currents: the role of passive membrane properties.Trends Neurosci. 17(4):161–166.

    Google Scholar 

  • Spruston N and Johnston D (1992) Perforated patch-clamp analysis of the passive membrane properties of three classes of hippocampal neurons.J. Neurophys. 67(3):508–529.

    Google Scholar 

  • Stafstrom CE, Schwindt PC, Chubb MC, and Crill WE (1985) Properties of persistent sodium conductance and calcium conductance of layer V neurons from cat sensorimotor cortex in vitro.J. Neurophysiology 53(1):153–170.

    Google Scholar 

  • Stein RB (1967) The frequency of nerve action potentials generated by applied currents.Proc. R. Soc. Lond. B 167:64–86.

    Google Scholar 

  • Steriade M, McCormick DA, and Sejnowski TJ (1993) Thalamocortical oscillations in the sleeping and aroused brain.Science 262:679–685.

    Google Scholar 

  • Stratford K, Mason A, Larkman A, Major G, and Jack JJB (1989) The modeling of pyramidal neurons in the visual cortex. In R Durbin, C Miall and G Mitchison, eds., The Computing Neuron, Addison-Wesley, London. Chapter 16, pp. 296–321.

    Google Scholar 

  • Stuart GJ and Sakmann B (1994) Active propagation of somatic action potentials into neocortical pyramidal cell dendrites.Nature 367:69–72.

    Google Scholar 

  • Traub RD and Miles R (1991) Multiple modes of neuronal population activity emerge after modifying specific synapses in a model of the CA3 region of the hippocampus.Ann. NY Acad. 627:277–290.

    Google Scholar 

  • Usher M, Stemmler M, Koch C, and Olami Z (1994) Network amplification of local fluctuations causes high spike rate variability, fractal firing patterns, and oscillatory local field potentials.Neural Computation 6:795–836.

    Google Scholar 

  • von der Malsburg C (1981) The correlation theory of brain function.Internal Report 81(2).

  • White E (1989) Cortical Circuits. Birkhaeuser, Boston.

    Google Scholar 

  • Wilson HR and Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons.Biophys. J. 12:1–24.

    Google Scholar 

  • Wolf H and Laurent G (1994) Rhythmic modulation of the responsiveness of locust sensory local interneurons by walking pattern generating networks.J. Neurophysiology 71(1):110–118.

    Google Scholar 

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Koch, C., Bernander, Ö. & Douglas, R.J. Do neurons have a voltage or a current threshold for action potential initiation?. J Comput Neurosci 2, 63–82 (1995). https://doi.org/10.1007/BF00962708

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