Abstract
At first glance, scientists seem to have needed a surprisingly long amount of time to find the physical correlate of thought. While the brain itself, as an organ, has long been considered the seat of the mind,Notable exceptions include Aristotle, who believed it to be a blood-cooling device (Gross 1995). it was only at the turn of the 20th century that Cajal and Golgi established the “neuron doctrine” —the hypothesis that the neuron is the fundamental functional unit of the brain.
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Notes
- 1.
Notable exceptions include Aristotle, who believed it to be a blood-cooling device (Gross 1995).
- 2.
Let us assume a neocortical pyramidal cell has a total area of \(S = 3 \times 10^4\,\upmu \)m\(^2\) and a volume of \(V = 10^4\,\upmu \)m\(^3\), with a specific membrane capacitance of \(c = 1\,\upmu \)F/cm\(^2\) and a \({\mathrm {Na}^+} \) concentration of \([{\mathrm {Na}^+} ] = 3 \times 10^7\) ions/\(\upmu \)m\(^3\). The number of ions required for an increase in membrane potential by 10 mV is then \(N = cSU/e \approx 2 \times 10^7\), as compared to the total number of ions in the cell \(N_\mathrm {tot} = [{\mathrm {Na}^+} ] V = 3 \times 10^{11}\). This is only an approximate calculation, as estimates of cell sizes vary considerably, see e.g. Ambros-Ingerson and Holmes (2005).
- 3.
The quintessential question in the search for the origin of life is how this gradient could have appeared in early cell-like structures. Furthermore, while in prokaryotic cells, ATP is synthesized at the plasma membrane, in eukaryotes this process is taken over by specialized organelles—mitochondria and plastids. It has been argued that it was the acquisition of these organelles by early prokaryotes that enabled the evolution of complex life. For an excellent discussion on these topics, we refer to Lane and Martin (2010, 2012).
- 4.
Actually, an electrochemical gradient would exist even without the \({\mathrm {Na}^+} \)–\({\mathrm {K}^+} \) pump, due to the high concentration of organic compounds inside the cell. These, in turn, cause a high intracellular osmolarity, which would force water to move into the cell by osmosis. The \({\mathrm {Na}^+} \)–\({\mathrm {K}^+} \) pump counters this potentially destructive effect by increasing the net extracellular concentration of inorganic ions.
- 5.
For ions of higher valence, such as \({\mathrm {Ca}^{++}} \), extensions to the GHK equation exist—see, e.g., Pickard (1976). However, since during resting conditions, both the permeability and the concentration of \({\mathrm {Ca}^{++}} \) ions is comparatively low, calcium does not play a significant role in defining \({V_\mathrm {rest}} \).
- 6.
Permeability and conductance are closely related, but not equivalent Channel conductance is, in particular, strongly voltage dependent. For a detailed discussion, see, e.g., Schultz et al. (1996).
- 7.
To get a feeling of the relevant variables, a numerical example is in order. For that, we consider measurements of the squid giant axon from Hodgkin (1958). The values given by Hodgkin read: \(P_{\mathrm {Na}^+} =1\), \(P_{\mathrm {K}^+} =100\), \(P_{\mathrm {Cl}^-} =10\) (permeabilities are given relative to \(P_{\mathrm {Na}^+} \)), \([{\mathrm {K}^+} ]_\mathrm {ext}=20\), \([{\mathrm {K}^+} ]_\mathrm {int}=200\), \([{\mathrm {Na}^+} ]_\mathrm {ext}=440\), \([{\mathrm {Na}^+} ]_\mathrm {int}=50\), \([{\mathrm {Cl}^-} ]_\mathrm {ext}=540\), \([{\mathrm {Cl}^-} ]_\mathrm {int}=40\) (ion concentrations given in mmol/l). At a temperature of 37 \(^\circ \)C, the Nernst potentials then read \(E_{{\mathrm {Na}^+}}=58.1\), \(E_{{\mathrm {K}^+}}=-61.5\) and \(E_{{\mathrm {Cl}^-}}=-69.6\), with the equilibrium membrane potential lying at \({E_\mathrm {l}} =-58.6\) (potentials given in mV).
- 8.
This is, of course, not absolutely true, since mechanisms such as local ion depletion, neurotransmitter diffusion or electrical crosstalk do enable additional communication pathways between neurons. Furthermore, electrical synapses (see Sect. 2.1.3) can also create a continuous link between membrane potentials.
- 9.
Modern extensions of the original model mainly include the addition of other types of ion channels and the morphology of neural cells.
- 10.
We have to stress that the Hodgkin–Huxley model is purely phenomenological and that the “gates” referenced multiple time in the text are only a mechanistic interpretation of the integer exponents in the gating variable equations. This is, however, quite close to reality: voltage-sensitive transmembrane proteins have, indeed, multiple identical compartments that change their conformation depending on the membrane potential. The voltage dependence of channel protein conformations is still the subject of intensive research, see, e.g., Long et al. (2007) for recent results on the structure of voltage-gated \({\mathrm {K}^+} \) channels.
- 11.
- 12.
Note that already in the deactivated state, individual n gates have a significant probability (\(p \approx 0.3\)) of being open. However, for a channel to be permeable, all gates have to be open at the same time, the probability of which scales with \(p^4\) and is therefore much lower.
- 13.
See also Kistler et al. (1997) for a similar discussion related to a different type of simplified neuron model (the spike-response model).
- 14.
More precisely, where the axon of the presynaptic cell touches a dendrite of the postsynaptic cell. See Sect. 2.1.4 for more details on neuron morphology and its functional consequences.
- 15.
Which stands for “postsynaptic potential” and is therefore a rather unfortunate acronym for something representing a temporary change in the latter.
- 16.
Coincidentally, the abbreviation “PSC” can stand for either postsynaptic current (generated by the influx of ions through the ligand-gated ion channels) or postsynaptic conductance. The reader is therefore encouraged to pay particular attention to the context in which this acronym appears.
- 17.
There has been quite some historical controversy surrounding the precise wording of Dale’s law. It concerns the ambiguity of the original statement from 1954 about whether one neuron may release multiple types of neurotransmitters at its terminals (Eccles et al. 1954). A revised version that is in compliance with today’s knowledge has been formulated by Eccles in 1976: “I proposed that Dale’s Principle be defined as stating that at all the axonal branches of a neurone, there was liberation of the same transmitter substance or substances” (Eccles 1976).
- 18.
As we have seen in Sect. 2.1.2, active (voltage-gated) channels on the membrane are a major component of membrane dynamics, even in the subthreshold regime. Dendrites also become thicker as they join and approach the soma, with the soma diameter being many times larger than that of distal dendrites. Finally, transmembrane currents do not sum up linearly, as the transmembrane proteins are not ideal resistors.
- 19.
For a dendrite with a radius of 1 \(\upmu \)m, we can find typical values of \({r_\mathrm {l}} = 3 \times 10^5{\Omega / \upmu \mathrm{m}}\), \({r_\mathrm {m}} = 5 \times 10^{11} {\Omega \upmu \mathrm{m}}\) and \({c_\mathrm {m}} = 5 \times 10^{-14}\mathrm{F} / \upmu \mathrm{m}\). The corresponding electrotonic length scale and membrane time constant are \({\lambda _\mathrm {m}} = 1.2\, \mathrm{mm}\) and \({\tau _\mathrm {m}} = 25\, \mathrm{ms}\).
- 20.
It is noteworthy that myelination can also be found in some older taxa as a result of convergent evolution. While not morphologically identical, invertebrate myelin sheaths serve the same functional purpose as in vertebrates.
- 21.
It is important to remember that this, too, is a simplification and does not hold without exception. While most neocortical neurons appear to conform to this assumption, networks exist—e.g., in the elephantnose fish—in which different shapes of action potentials have been measured (Sugawara et al. 1999) and hypothesized to play a functional role (Mohr et al. 2003a, b).
- 22.
As an example, we mention the Poincaré–Bendixson theorem, which makes a statement about the periodicity of orbits (limit cycles) for two-dimensional dynamical systems. In particular, it forbids the existence of chaotic behavior such as strange attractors. This clearly does not hold for higher-dimensional phase spaces, such as the Lorenz system with its well-known “butterfly attractor”.
- 23.
- 24.
This is the standard textbook definition, but in this formulation of the model, the “from below” can be omitted, since the equations prevent the membrane potential from ever lying above the threshold.
- 25.
What works well in theory may not be equally unproblematic in practice. Expressions which may converge to finite values, but contain terms that diverge in the required limit, are notoriously problematic in software implementations. Neural simulation software handles such problems with varying degrees of success: NEST 2.1.1, for example, returns an error, while Neuron 7.1 returns a warning. Therefore, when such a limit is required (such as for the L23 model fit in Sect. 5.3), particular care needs to be taken. For hardware implementations, such terms become even more problematic, due to limited parameter precision (see Sect. A.2.2). On the HICANN chip, this issue is solved by having the exponential term implemented in a separate circuit that can be effectively decoupled from the cell membrane (see Sect. 3.3.1).
- 26.
In the spirit of the Hodgkin–Huxley model of neuron/membrane dynamics.
- 27.
Incidentally, this function is identical to the PSP shapes derived in Sect. 4.2 (Eqs. 4.39 and 4.59). In order to avoid any confusion, we note explicitly that Eq. 2.48 represents a phenomenological model of PSCs, whereas Eqs. 4.39 and 4.59 represents the shape of a PSP, i.e., the analytical solution of the LIF equation driven by a single, exponentially shaped PSC.
- 28.
We point out again that, depending on the context, the abbreviation PSC may refer to either a postsynaptic current or a postsynaptic conductance.
- 29.
In PyNN, for example, neuron models implicitly characterize their synapse dynamics. This is accounted for by the typical naming of neuron models, such as, e.g., IF_cond_alpha or aEIF_curr_exp.
- 30.
Sometimes long-term and short-term plasticity are abbreviated as LTP and STP, respectively. This can be easily confused with long-term and short-term potentiation and must be inferred from the context, if necessary. Here, we use “P” as an abbreviation for potentiation and do not abbreviate short-term- and long-term plasticity.
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Petrovici, M.A. (2016). Introduction: From Biological Experiments to Mathematical Models. In: Form Versus Function: Theory and Models for Neuronal Substrates . Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-39552-4_2
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