Spike response model

In the SRM, at each moment in time t, a spike can be generated stochastically with instantaneous stochastic intensity or 'escape function'

: $\backslash rho(t)\; =\; f(V(t)-\backslash vartheta(t))$

that depends on the momentary difference between the membrane voltage {{math|V(t)}} and the dynamic threshold $\backslash vartheta(t)$.

The membrane voltage {{math|V(t)}} at time t is given by

: $V(t)=\; \backslash sum\_f\; \backslash eta(t-t^f)\; +\; \backslash int\_0^\backslash infty\; \backslash kappa(s)\; I(t-s)\; \backslash ,\; ds\; +\; V\_\backslash mathrm\{rest\}$

where {{math|t^f}} is the firing time of spike number f of the neuron, {{math|V_rest}} is the resting voltage in the absence of input, {{math|I(t-s)}} is the input current at time t − s and $\backslash kappa(s)$ is a linear filter (also called kernel) that describes the contribution of an input current pulse at time t − s to the voltage at time t. The contributions to the voltage caused by a spike at time $t^f$ are described by the refractory kernel $\backslash eta(t-t^f)$. In particular,

$\backslash eta(t-t^f)$ describes the time course of the action potential starting at time $t^f$ as well as the spike-afterpotential.

The dynamic threshold $\backslash vartheta(t)$ is given by

: $\backslash vartheta(t)=\; \backslash vartheta\_0\; +\; \backslash sum\_f\; \backslash theta\_1(t-t^\{f\})$

where $\backslash vartheta\_0$ is the firing threshold of an inactive neuron and $\backslash theta\_1(t-t^f)$ describes the increase of the threshold after a spike at time $t^f$. In case of a fixed threshold [i.e., $\backslash theta\_1(t-t^f)$=0], the refractory kernel $\backslash eta(t-t^f)$ should include only the spike-afterpotential, but not the shape of the spike itself.

A common choice for the 'escape rate' $f$ (that is consistent with biological data) is

: $f(V-\backslash vartheta)\; =\; \backslash frac\{1\}\{\backslash tau\_0\}\; \backslash exp[\backslash beta(V-\backslash vartheta)]$

where $\backslash tau\_0$ is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and $\backslash beta$ is a sharpness parameter. For $\backslash beta\backslash to\backslash infty$ the threshold becomes sharp and spike firing occurs deterministically at the moment when the membrane potential hits the threshold from below. The sharpness value found in experiments is $1/\backslash beta\backslash approx\; 4mV$ which that neuronal firing becomes non-neglibable as soon the membrane potential is a few mV below the formal firing threshold. The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook Neuronal Dynamics.

In a network of N SRM neurons $1\backslash le\; i\; \backslash le\; N$, the membrane voltage of neuron $i$ is given by

: $V\_i(t)=\; \backslash sum\_f\; \backslash eta\_i(t-t\_i^\{f\})\; +\; \backslash sum\_\{j=1\}^N\; w\_\{ij\}\; \backslash sum\_\{f\text{'}\}\backslash varepsilon\_\{ij\}(t-t\_j^\{f\text{'}\})\; +\; V\_\backslash mathrm\{rest\}$

where $t\_j^\{f\text{'}\}$ are the firing times of neuron j (i.e., its spike train), and $\backslash eta\_i(t-t^f\_i)$ describes the time course of the spike and the spike after-potential for neuron i, $w\_\{ij\}$ and $\backslash varepsilon\_\{ij\}(t-t\_j^\{f\text{'}\})$ describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike $t\_j^\{f\text{'}\}$ of the presynaptic neuron j. The time course $\backslash varepsilon\_\{ij\}(s)$ of the PSP results from the convolution of the postsynaptic current $I(t)$ caused by the arrival of a presynaptic spike from neuron j.

For simulations, the SRM is usually implemented in discrete time.{{Cite journal|last=Gerstner|first=Wulfram|date=2000-01-01|title=Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking|url=https://doi.org/10.1162/089976600300015899|journal=Neural Computation|volume=12|issue=1|pages=43–89|doi=10.1162/089976600300015899|pmid=10636933|s2cid=7832768|issn=0899-7667}} In time step $t\_n$ of duration $\backslash Delta\; t$, a spike is generated with probability

: $P\_F(t\_n)\; =\; F(V(t\_n)-\backslash vartheta(t\_n))$

that depends on the momentary difference between the membrane voltage {{math|V}} and the dynamic threshold $\backslash vartheta$. The function F is often taken as a standard sigmoidal $F(x)\; =\; 0.5[1\; +\; \backslash tanh(\backslash gamma\; x)]${{Cite journal|last=GERSTNER|first=Wulfram|date=1991|title=Associative memory in a network of biological neurons|url=https://papers.nips.cc/paper/371-associative-memory-in-a-network-of-biological-neurons.pdf |journal=Advances in Neural Information Processing Systems|volume=3|pages=84–90}} with steepness parameter $\backslash gamma$. But the functional form of F can also be calculated from the stochastic intensity $f$ in continuous time as $F(y\_n)\backslash approx\; 1\; -\; \backslash exp[y\_n\; \backslash ,\; \backslash Delta\; t]$ where $y\_n\; =\; V(t\_n)-\backslash vartheta(t\_n)$ is the distance to threshold.

The membrane voltage $V(t\_n)$ in discrete time is given by

: $V(t\_n)\; =\; \backslash sum\_f\; \backslash eta(t\_n-t^\{f\})\; +\; \backslash sum\_\{m=1\}^\backslash infty\; \backslash kappa(m\backslash ,\backslash Delta\; t)\; I(t\_n-m\backslash ,\backslash Delta\; t)+\; V\_\backslash mathrm\{rest\}$

where {{math|t^f}} is the discretized firing time of the neuron, {{math|V_rest}} is the resting voltage in the absence of input, and $I(t\_k)$ is the input current at time $t\_k$ (integrated over one time step). The input filter $\backslash kappa(s)$ and the spike-afterpotential $\backslash eta(s)$ are defined as in the case of the SRM in continuous time.

For networks of SRM neurons in discrete time we define the spike train of neuron j as a sequence of zeros and ones, $\backslash \{X\_j(t\_m)\backslash in\; \backslash \{0,1\backslash \};\; m=1,2,3,\; \backslash dots\; \backslash \}$

and rewrite the membrane potential as

: $V\_i(t\_n)\; =\; \backslash sum\_m\backslash eta\_i(t\_n-t\_m)\; X\_i(t\_m)\; +\; \backslash sum\_j\; w\_\{ij\}\; \backslash sum\_m\; \backslash varepsilon\_\{ij\}(t\_n-t\_m)\; X\_j(t\_m)\; +\; V\_\backslash mathrm\{rest\}$

In this notation, the refractory kernel $\backslash kappa(s)$ and the PSP shape $\backslash varepsilon\_\{ij\}(s)$ can be interpreted as linear response filters applied to the binary spike trains $X\_j$.

Since the formulation as SRM provides an explicit expression for the membrane voltage (without the detour via a differential equations), SRMs have been the dominant mathematical model in a formal theory of computation with spiking neurons.{{Cite book|url=https://www.worldcat.org/oclc/42856203|title=Pulsed neural networks|date=1999|publisher=MIT Press|others=Maass, Wolfgang, 1949 August 21-, Bishop, Christopher M.|isbn=978-0-262-27876-8|location=Cambridge, Mass.|oclc=42856203}}{{Cite journal|last1=Bohte|first1=Sander M.|last2=Kok|first2=Joost N.|last3=La Poutré|first3=Han|date=2002-10-01|title=Error-backpropagation in temporally encoded networks of spiking neurons|url=http://www.sciencedirect.com/science/article/pii/S0925231201006580|journal=Neurocomputing|language=en|volume=48|issue=1|pages=17–37|doi=10.1016/S0925-2312(01)00658-0|issn=0925-2312}}{{Cite journal|last=Maass|first=Wolfgang|date=1997-02-01|title=Fast Sigmoidal Networks via Spiking Neurons|url=https://doi.org/10.1162/neco.1997.9.2.279|journal=Neural Computation|volume=9|issue=2|pages=279–304|doi=10.1162/neco.1997.9.2.279|pmid=9117904|s2cid=34984298 |issn=0899-7667|url-access=subscription}}

The SRM with dynamic threshold has been used to predict the firing time of cortical neurons with a precision of a few milliseconds. Neurons were stimulated, via current injection, with time-dependent currents of different means and variance while the membrane voltage was recorded. The reliability of predicted spikes was close to the intrinsic reliability when the same time-dependent current was repeated several times. Moreover, extracting the shape of the filters $\backslash kappa(s)$ and $\backslash eta(s)$ directly from the experimental data revealed that adaptation extends over time scales from tens of milliseconds to tens of seconds.{{Cite journal|last1=Pozzorini|first1=Christian|last2=Naud|first2=Richard|last3=Mensi|first3=Skander|last4=Gerstner|first4=Wulfram|date=July 2013|title=Temporal whitening by power-law adaptation in neocortical neurons|url=https://www.nature.com/articles/nn.3431|journal=Nature Neuroscience|language=en|volume=16|issue=7|pages=942–948|doi=10.1038/nn.3431|pmid=23749146|s2cid=1873019|issn=1546-1726}}{{Cite journal|last1=Mensi|first1=Skander|last2=Naud|first2=Richard|last3=Pozzorini|first3=Christian|last4=Avermann|first4=Michael|last5=Petersen|first5=Carl C. H.|last6=Gerstner|first6=Wulfram|date=2011-12-07|title=Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms|url=https://journals.physiology.org/doi/full/10.1152/jn.00408.2011|journal=Journal of Neurophysiology|volume=107|issue=6|pages=1756–1775|doi=10.1152/jn.00408.2011|pmid=22157113|issn=0022-3077}}{{Cite journal |last1=Haslinger |first1=Robert |last2=Pipa |first2=Gordon |last3=Lima |first3=Bruss |last4=Singer |first4=Wolf |last5=Brown |first5=Emery N. |last6=Neuenschwander |first6=Sergio |date=2012-07-03 |title=Context Matters: The Illusive Simplicity of Macaque V1 Receptive Fields |journal=PLOS ONE |language=en |volume=7 |issue=7 |pages=e39699 |doi=10.1371/journal.pone.0039699 |doi-access=free |issn=1932-6203 |pmc=3389039 |pmid=22802940|bibcode=2012PLoSO...739699H }} Thanks to the convexity properties of the likelihood in Generalized Linear Models, parameter extraction is efficient.{{Cite journal|last1=Pozzorini|first1=Christian|last2=Mensi|first2=Skander|last3=Hagens|first3=Olivier|last4=Naud|first4=Richard|last5=Koch|first5=Christof|last6=Gerstner|first6=Wulfram|date=2015-06-17|title=Automated High-Throughput Characterization of Single Neurons by Means of Simplified Spiking Models|journal=PLOS Computational Biology|language=en|volume=11|issue=6|pages=e1004275|doi=10.1371/journal.pcbi.1004275|issn=1553-7358|pmc=4470831|pmid=26083597|bibcode=2015PLSCB..11E4275P |doi-access=free }}

SRM0 neurons have been used to construct an associative memory in a network of spiking neurons. The SRM network which stored a finite number of stationary patterns as attractors using a Hopfield-type connectivity matrix{{Cite journal|last=Hopfield|first=J. J.|date=1982-04-01|title=Neural networks and physical systems with emergent collective computational abilities|journal=Proceedings of the National Academy of Sciences|language=en|volume=79|issue=8|pages=2554–2558|doi=10.1073/pnas.79.8.2554|issn=0027-8424|pmc=346238|pmid=6953413|bibcode=1982PNAS...79.2554H|doi-access=free }} was one of the first examples of attractor networks with spiking neurons.{{Cite journal|last1=Amit|first1=Daniel J.|last2=Tsodyks|first2=M. V.|date=1991-01-01|title=Quantitative study of attractor neural network retrieving at low spike rates: I. substrate—spikes, rates and neuronal gain|url=https://doi.org/10.1088/0954-898X_2_3_003|journal=Network: Computation in Neural Systems|volume=2|issue=3|pages=259–273|doi=10.1088/0954-898X_2_3_003|issn=0954-898X|url-access=subscription}}{{Cite journal|last=Amit|first=D.|date=1997-04-01|title=Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex|journal=Cerebral Cortex|volume=7|issue=3|pages=237–252|doi=10.1093/cercor/7.3.237|pmid=9143444|issn=1460-2199|doi-access=free}}{{Cite journal|last1=Brader|first1=Joseph M.|last2=Senn|first2=Walter|last3=Fusi|first3=Stefano|date=2007-09-20|title=Learning Real-World Stimuli in a Neural Network with Spike-Driven Synaptic Dynamics|url=https://doi.org/10.1162/neco.2007.19.11.2881|journal=Neural Computation|volume=19|issue=11|pages=2881–2912|doi=10.1162/neco.2007.19.11.2881|pmid=17883345|s2cid=5933554|issn=0899-7667}}

For SRM neurons, an important variable characterizing the internal state of the neuron is the time since the last spike (or 'age' of the neuron) which enters into the refractory kernel $\backslash eta(s)$. The population activity equations for SRM neurons can be formulated alternatively either as integral equations,{{Cite journal|last1=Schwalger|first1=Tilo|last2=Deger|first2=Moritz|last3=Gerstner|first3=Wulfram|date=2017-04-19|editor-last=Graham|editor-first=Lyle J.|title=Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size|journal=PLOS Computational Biology|language=en|volume=13|issue=4|pages=e1005507|doi=10.1371/journal.pcbi.1005507|issn=1553-7358|pmc=5415267|pmid=28422957|arxiv=1611.00294|bibcode=2017PLSCB..13E5507S |doi-access=free }} or as partial differential equations for the 'refractory density'. Because the refractory kernel may include a time scale slower than that of the membrane potential, the population equations for SRM neurons provide powerful alternatives{{Cite journal|last1=Naud|first1=Richard|last2=Gerstner|first2=Wulfram|date=2012-10-04|title=Coding and Decoding with Adapting Neurons: A Population Approach to the Peri-Stimulus Time Histogram|journal=PLOS Computational Biology|language=en|volume=8|issue=10|pages=e1002711|doi=10.1371/journal.pcbi.1002711|issn=1553-7358|pmc=3464223|pmid=23055914|bibcode=2012PLSCB...8E2711N |doi-access=free }}{{Cite journal|last1=Schwalger|first1=Tilo|last2=Chizhov|first2=Anton V|date=2019-10-01|title=Mind the last spike — firing rate models for mesoscopic populations of spiking neurons|url=http://www.sciencedirect.com/science/article/pii/S095943881930039X|journal=Current Opinion in Neurobiology|series=Computational Neuroscience|language=en|volume=58|pages=155–166|doi=10.1016/j.conb.2019.08.003|pmid=31590003|arxiv=1909.10007|s2cid=202719555|issn=0959-4388}} to the more broadly used partial differential equations for the 'membrane potential density'.{{Cite journal|last=Treves|first=Alessandro|date=January 1993|title=Mean-field analysis of neuronal spike dynamics|url=https://www.tandfonline.com/doi/full/10.1088/0954-898X_4_3_002|journal=Network: Computation in Neural Systems|language=en|volume=4|issue=3|pages=259–284|doi=10.1088/0954-898X_4_3_002|issn=0954-898X|url-access=subscription}}{{Cite journal|last=Brunel|first=Nicolas|date=2000-05-01|title=Dynamics of Sparsely Connected Networks of Excitatory and Inhibitory Spiking Neurons|url=https://doi.org/10.1023/A:1008925309027|journal=Journal of Computational Neuroscience|language=en|volume=8|issue=3|pages=183–208|doi=10.1023/A:1008925309027|pmid=10809012|s2cid=1849650|issn=1573-6873|url-access=subscription}} Reviews of the population activity equation based on refractory densities can be found in as well in Chapter 14 of the textbook Neuronal Dynamics.

SRMs are useful to understand theories of neural coding. A network SRM neurons has stored attractors that form reliable spatio-temporal spike patterns (also known as synfire chains{{Cite journal|last=Abeles|first=Moshe|date=2009|title=Synfire chains|journal=Scholarpedia|language=en|volume=4|issue=7|pages=1441|doi=10.4249/scholarpedia.1441|bibcode=2009SchpJ...4.1441A|issn=1941-6016|doi-access=free}}) example of temporal coding for stationary inputs. Moreover, the population activity equations for SRM exhibit temporally precise transients after a stimulus switch, indicating reliable spike firing.

The Spike Response Model has been introduced in a series of papers between 1991 and 2000.{{Cite journal|last=Gerstner|first=Wulfram|date=1995-01-01|title=Time structure of the activity in neural network models|url=https://link.aps.org/doi/10.1103/PhysRevE.51.738|journal=Physical Review E|volume=51|issue=1|pages=738–758|doi=10.1103/PhysRevE.51.738|pmid=9962697|bibcode=1995PhRvE..51..738G}} The name Spike Response Model probably appeared for the first time in 1993. Some papers used exclusively the deterministic limit with a hard threshold others the soft threshold with escape noise. Precursors of the Spike Response Model are the integrate-and-fire model introduced by Lapicque in 1907 as well as models used in auditory neuroscience.{{Cite journal|last=Weiss|first=Thomas F.|date=1966-11-01|title=A model of the peripheral auditory system|url=https://doi.org/10.1007/BF00290252|journal=Kybernetik|language=en|volume=3|issue=4|pages=153–175|doi=10.1007/BF00290252|pmid=5982096|s2cid=30861035 |issn=1432-0770|url-access=subscription}}

An important variant of the model is SRM0 which is related to time-dependent nonlinear renewal theory. The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel $\backslash eta(s)$ there is no summation sign over past spikes: only the most recent spike matters. The model SRM0 is closely related to the inhomogeneous Markov interval process{{Cite journal|last1=Kass|first1=Robert E.|last2=Ventura|first2=Valérie|date=2001-08-01|title=A Spike-Train Probability Model|url=https://www.mitpressjournals.org/doi/abs/10.1162/08997660152469314|journal=Neural Computation|language=en|volume=13|issue=8|pages=1713–1720|doi=10.1162/08997660152469314|pmid=11506667|s2cid=9909632 |issn=0899-7667|url-access=subscription}} and to age-dependent models of refractoriness.{{Cite journal|last1=Johnson|first1=Don H.|last2=Swami|first2=Ananthram|date=1983-08-01|title=The transmission of signals by auditory-nerve fiber discharge patterns|url=https://asa.scitation.org/doi/10.1121/1.389815|journal=The Journal of the Acoustical Society of America|volume=74|issue=2|pages=493–501|doi=10.1121/1.389815|pmid=6311884|bibcode=1983ASAJ...74..493J |issn=0001-4966|url-access=subscription}}{{Cite journal|last1=Berry|first1=Michael J.|last2=Meister|first2=Markus|date=1998-03-15|title=Refractoriness and Neural Precision|url= |journal=The Journal of Neuroscience|language=en|volume=18|issue=6|pages=2200–2211|doi=10.1523/JNEUROSCI.18-06-02200.1998|issn=0270-6474|pmc=6792934|pmid=9482804}}{{Cite journal|last1=Gaumond|first1=R P|last2=Molnar|first2=C E|last3=Kim|first3=D O|date=September 1982|title=Stimulus and recovery dependence of cat cochlear nerve fiber spike discharge probability.|url=https://www.physiology.org/doi/10.1152/jn.1982.48.3.856|journal=Journal of Neurophysiology|language=en|volume=48|issue=3|pages=856–873|doi=10.1152/jn.1982.48.3.856|pmid=6290620|issn=0022-3077|url-access=subscription}}

The equations of the SRM as introduced above are equivalent to Generalized Linear Models in neuroscience (GLM). In the neuroscience, GLMs have been introduced as an extension of the Linear-Nonlinear-Poisson model (LNP) by adding self-interaction of an output spike with the internal state of the neuron (therefore also called 'Recursive LNP'). The self-interaction is equivalent to the kernel $\backslash eta(s)$ of the SRM. The GLM framework enables to formulate a maximum likelihood approach{{Cite journal|last=Brillinger|first=D. R.|date=1988-08-01|title=Maximum likelihood analysis of spike trains of interacting nerve cells|url=https://doi.org/10.1007/BF00318010|journal=Biological Cybernetics|language=en|volume=59|issue=3|pages=189–200|doi=10.1007/BF00318010|pmid=3179344|s2cid=10164135|issn=1432-0770|url-access=subscription}} applied to the [https://neuronaldynamics.epfl.ch/online/Ch9.S2.html likelihood of an observed spike train] under the assumption that an SRM could have generated the spike train. Despite the mathematical equivalence there is a conceptual difference in interpretation: in the SRM the variable V is interpreted as membrane voltage whereas in the recursive LNP it is a 'hidden' variable to which no meaning is assigned. The SRM interpretation is useful if measurements of subthreshold voltage are available whereas the recursive LNP is useful in systems neuroscience where spikes (in response to sensory stimulation) are recorded extracellulary without access to the subthreshold voltage.

A leaky integrate-and-fire neuron with spike-triggered adaptation has a subthreshold membrane potential generated by the following differential equations

: $\backslash tau\_\backslash mathrm\{m\}\; \backslash frac\{d\; V\; (t)\}\{d\; t\}\; =\; R\; I(t)-\; [V\; (t)\; -\; E\_\backslash mathrm\{rest\}\; ]-\; R\; \backslash sum\_k\; w\_k$

: $\backslash tau\_k\; \backslash frac\{d\; w\_k\; (t)\}\{d\; t\}\; =\; -\; w\_k\; +\; b\_k\; \backslash tau\_k\; \backslash sum\_f\; \backslash delta\; (t-t^f)$

where $\backslash tau\_m$ is the membrane time constant and {{math|w_k}} is an adaptation current number, with index k, {{math|E_rest}} is the resting potential and {{math|t^f}} is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value {{math|V_r}} below the firing threshold. Integration of the linear differential equations gives a formula identical to the voltage equation of the SRM. However, in this case, the refractory kernel $\backslash eta(s)$ does not include the spike shape but only the spike-afterpotential. In the absence of adaptation currents, we retrieve the standard LIF model which is equivalent to a refractory kernel $\backslash eta(s)$ that decays exponentially with the membrane time constant $\backslash tau\_m$.

• [https://neuronaldynamics.epfl.ch/online/Ch6.S4.html Spike Response Model], Chapter 6.4 of the textbook Neuronal Dynamics
• [https://neuronaldynamics.epfl.ch/online/Ch9.html 'soft threshold' and escape noise], Chapter 9 of the textbook Neuronal Dynamics
• [https://neuronaldynamics.epfl.ch/online/Ch14.html Quasi-Renewal Theory] Chapter 14 of the textbook Neuronal Dynamics.
• [http://scholarpedia.org/article/Spike-response_model Spike Response Model], from Scholarpedia{{Cite journal|last=Gerstner|first=Wulfram|date=2008|title=Spike-response model|journal=Scholarpedia|language=en|volume=3|issue=12|pages=1343|doi=10.4249/scholarpedia.1343|bibcode=2008SchpJ...3.1343G |issn=1941-6016|doi-access=free}}

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