in Neuronal Population with Phase Response

(1)

where $\frac{\varepsilon }{N}{\displaystyle \sum_n\alpha \left(t-{t}_k^n\right)}$ is the input to the jth neuronal oscillator from the kth neuronal oscillator, ε is weak coupling constant, N is the total number of neuronal oscillators, t _n^kis the nth firing time of the kth neuronal oscillator, α(t) is a causal coupling function. θ _jis the phase of jth neuronal oscillator, ω is the natural frequency of a neuronal oscillator, Z(θ) is a phase response curve of a neuronal oscillator, c sin(ω ₀ t) is an external periodic force with a strength c and frequency ω ₀.

We assume that the mutual interaction shift the frequency of the mean phase of these oscillators by εΩ from the natural frequency ω, and define the relative phase ψ _j = θ _j − (ω + εΩ)t. The relative phase ψ _jchanges slowly compared with and will hardly change during the oscillation period θ _j. Therefore, we substituteψ _jinto Eq.(1), and average Eq. (1) over one period keeping ψ _jconstant, so the relative phase ψ _jobeys the following equation:

$\begin{array}{ll}\frac{d{\psi}_j}{dt} & =-\varepsilon \Omega +\frac{\varepsilon }{NT}{\displaystyle {\int}_0^TZ\left({\psi}_j+\left(\omega +\varepsilon \Omega \right)t\right)}{\displaystyle \sum_{k=1}^N{\displaystyle \sum_n\alpha \left(t-{t}_k^n\right)}}\\ & \quad+ c \sin \left(\omega t\right)Z\left({\psi}_j+\left(\omega +\varepsilon \Omega \right)t\right) \end{array}$

(2)

In order to investigate the dynamic response of neural population, we introduce complex order parameters describing synchronized phenomenon in the neuronal population

${\mathrm{Re}}^{i\psi}=\frac{1}{N}{\displaystyle \sum_{j=1}^N{e}^{i{\psi}_j}}$

(3)

where R is the amplitude of the order parameters describing the degree of synchronization of neuronal oscillators, 0 ≤ R ≤ 1, the bigger R show that synchronous activity is stronger, ψ is the average phase of the neuronal population.

3 Result

We investigate the response property of neuronal population to external periodic stimulus. The phase sensitivity function Z(θ) is considered as a sinusoidal sensitivity function sin(θ) as in Ref. [15]. As synaptic time constant τ is smaller, the neuronal population quickly synchronized in-phase (Fig. 1a); but with τ increased, periodic synchronization occurred, and as the synaptic time constant is larger, the synchronization become weaker (Fig. 1b, c); even more, synchronized activity can be lost (Fig. 1d). This shows that synaptic time constant is an important condition under which the global neural network synchronized.