Finding the external fields $h_{i}$ and the coupling parameters $J_{ij}$ can fit the same means and pairwise covariance as the experiment data, using the equations

$\left<s_{i}\right>_{Ising} = \left<s_{i}\right>_{data}$

$\left<s_{i}s_{j}\right>_{Ising} = \left<s_{i}s_{j}\right>_{data}$.

We can solve the equations by iterations $h_{i}=h_{i}+\delta h_{i}$ and $J_{ij}=J_{ij}+\delta J_{ij}$ then make changes

$\delta h_{i}=\eta \{\left<s_{i}\right>_{data} - \left<s_{i}\right>_{Ising} \}$

$\delta J_{ij}=\eta \{\left<s_{i}s_{j}\right>_{data} - \left<s_{i}s_{j}\right>_{Ising} \}$.

We use the data ACV_slice5 (6 neurons) as our sample and set $\eta = 0.1$. The cost function is

${(\left<s_{i}\right>_{data} - \left<s_{i}\right>_{Ising})}^{2} \leq 0.00000001$ && ${(\left<s_{i}s_{j}\right>_{data} - \left<s_{i}s_{j}\right>_{Ising})}^{2} \leq 0.00000001$.

The experiment result and simulation result see figure.

If the number of neurons is larger than 30, it is very slow to find the suitable $h_{i}$ and $J_{ij}$ for the good fitting.

The mean field theory (nMF):

$\mathbf{J}^\text{nMF}=-\mathbf{C}^{-1}, (i \neq j)$,

$h_{i}^\text{nMF}=\tanh^{-1}\left< s_{i} \right>-\sum_{j=1}^{N} J{ij}^\text{nMF}\left< s_{j} \right>$ .

The independent-pair theory (IP):

$J_{ij}^\text{pair}=0.25*\ln\left[ \frac{(1+m_{i}+m_{j}+C_{ij}^{*}) (1-m_{i}-m_{j}+C_{ij}^{*})} {(1-m_{i}+m_{j}-C_{ij}^{*})(1+m_{i}-m_{j}-C_{ij}^{*})} \right]$, with $C_{ij}^{*}=C_{ij}+m_{i}m_{j} , (i \neq j)$

$h_{i}^\text{pair}=0.5*\ln\left( \frac{1+m_{i}} {1-m_{i}} \right)-\sum_{j=1}^{N} J_{ij}^\text{pair} m_{j}$.

The Thouless-Anderson-Palmer theory (TAP):

$\mathbf{C}^{-1}+J_{ij}^\text{TAP}+2(J_{ij}^\text{TAP})^{2}m_{i}m_{j}=0 , (i \neq j)$

$h_{i}^\text{TAP}=h_{i}^\text{nMF}-m_{i}\sum_{i=1}^{N} (J_{ij}^\text{TAP})^{2} (1-m_{j})^{2}$.

The Sessak and Monasson theory (SM):

$J_{ij}^\text{SM}=J_{ij}^\text{nMF}+J_{ij}^\text{pair}-\frac{C_{ij}}{(1-m_{i}^{2})(1-m_{j}^{2})-C_{ij}^{2}} , (i \neq j)$

$h_{i}^\text{SM}=0.5*\ln\left( \frac{1+m_{i}} {1-m_{i}} \right)-\sum_{j=1}^{N} J_{ij}^\text{SM} m_{j}$.

The coefficient of $R^{2}$ is:

$R^{2}=1-\frac{\sum_{ij} \left(J_{ij}^{\text{approx}}-J_{ij}^{\text{Exact}} \right)^{2} }{\sum_{ij} \left(J_{ij}^{\text{Exact}}-J_{ij}^{\text{$\overline{Exact}$}} \right)^{2}}$ , with

$\overline{J_{ij}^{\text{Exact}}}=\frac {\sum_{i \neq j} J_{ij}^{\text{Exact}}}{N(N-1)}$.

The rms error is: $\sqrt{\frac{1}{N(N-1)} \sum_{i \neq j} (J_{ij}^\text{approx}-J_{ij}^\text{Exact})^{2} }$.

We set the mean equal to zero and the variance of normal distribution of h and J from $\frac{1}{\sqrt{200}}$: $\frac{1}{\sqrt{150}}$: $\frac{1}{\sqrt{100}}$:$\frac{1}{\sqrt{50}}$:$\frac{1}{\sqrt{20}} \approx 0.07 : 0.08 : 0.1 : 0.14 : 0.22$.

We use the normal distribution of h and J to get the coefficients of $R^{2}$ and the RMSerror at T=1.