Measuring spike timing distance in the Hindmarsh–Rose neurons
Received: 9 May 2017 / Revised: 28 November 2017 / Accepted: 19 December 2017 / Published online: 27 December 2017
Ó Springer Science+Business Media B.V., part of Springer Nature 2017
In the present paper, a simple spike timing distance is deﬁned which can be used to measure the degree of synchronization
with the information only encoded in the precise timing of the spike trains. Via calculating the spike timing distance
deﬁned in this paper, the spike train similarity of uncoupled Hindmarsh–Rose neurons in bursting or spiking states with
different initial conditions is investigated and the results are compared with other spike train distance measures. Later, the
spike timing distance measure is applied to study the synchronization of coupled or common noise-stimulated neurons.
Counterintuitively, the addition of weak coupling or common noise doesn’t enhance the degree of synchronization
although after critical values, both of them can induce complete synchronizations. More interestingly, the common noise
plays opposite roles for weak and strong enough couplings. Finally, it should be noted that the measure deﬁned in this
paper can be extended to measure large neuronal ensembles and the lag synchronization.
Keywords Spike timing distance Á Synchronization Á Hindmarsh–Rose neuron Á Common noise
Understanding neural coding is an important part of
understanding informational neurodynamics in our brains.
For example, the rate codes play an important role in
identifying the regulation of the neuronal ﬁring trains (Guo
et al. 2016a, b). For temporal codes, two different types of
encoding, namely, the frequency of ﬁring and the exact
temporal occurrence of spikes, are reported in experimental
recordings (Rabinovich et al. 2006). Spike trains’ similarity
and dissimilarity can thus have two kinds of metrics, the
inter spike interval (ISI) distance (Kreuz et al. 2007) and
the spike distance (Kreuz et al. 2011, 2013). A measure by
introducing a kernel function was proposed by van Rossum
(2001), whereby the distance could interpolate between
coincidence detection and spike count difference via
changing the time constant, which gave rise to a continuous
prototype of spike train topology. A similar approach is
proposed recently by Rusu and Florian (2014), where they
show their max-metric and modulus-metric are particularly
suitable for measuring distances in spike trains where
information is encoded in the identity of bursts as unitary
events. For classiﬁcations of spike train distance, one can
also refer to Victor (2015) and references therein, where he
classiﬁed spike train metrics into embedding-based
(Houghton and Sen 2008; van Rossum 2001) and cost-
based (Victor and Purpura 1998) distances.
Synchronization (Pikovsky et al. 2003) as a collective
behavior has attracted much attention in neuroscience
partly due to its relation with many brain disorders, e.g.
schizophrenia, epilepsy, autism, Alzheimer’s disease, and
Parkinson’s disease, all of which are considered associated
with abnormal neural synchronizations (Uhlhaas and
Singer 2006). In neuronal ensembles, synchronizations
may be impacted by connection topologies (Bera et al.
2017; Majhi et al. 2016) and other systematic parameters,
e.g. time delay (Dhamala et al. 2004b; Sun et al. 2017; Zhu
et al. 2016), coupling strength (Dhamala et al. 2004a;
Ivanchenko et al. 2004), network size (Zhu et al. 2016), etc.
For coupled spiking neurons, the lag synchronization (LG)
can be measured by using the similarity function (Rosen-
blum et al. 1997), e.g. in the Rulkov map neurons (Zhu
et al. 2016) and in the Morris–Lecar neurons (Wang et al.
2013). However, it has been shown that LG does not occur
& Xianbin Liu
State Key Laboratory of Mechanics and Control of
Mechanical Structures, College of Aerospace Engineering,
Nanjing University of Aeronautics and Astronautics,
29 YuDao Street, Nanjing 210016, Jiangsu Province,
People’s Republic of China
Cognitive Neurodynamics (2018) 12:225–234