How to reduce thermal resistor noise in filter circuit?
The channel impedance is dominated by the electrode impedance shunted by the INA R. Typically EEG electrodes must be very low to reduce motion artifact noise and is about 50k / 50nF, thus the R &gt;&gt; 100k is insignificant.


$$e^2 = 4k_B\ T\ R\ Δf = 4 * 1.38x10^{-23} * 300K * 50kΩ * 300Hz \approx 0\  nV$$

where: k is Boltzmann’s constant 1.38*10^{–23} [Joule/K], T is temperature in Kelvin [K], R [Ω], Δf is frequency [Hz], e is Voltage (\$V_{rms}\$).


There is another 50 year old solution to this.  It was called Active Guarding, while others who also thought they invented it later, called it Active Electrode.  



  It guards the signal by driving the shielded cable to cancel the stray CM E-field rather than connected to CM ground.






Ref.    


shielding vs active electrodes   
Interference_reduction publication    
lowcost_active_electrode_improves_resolution     
Analog Devices Appication Note    
Linear Technology Notes 


The 50 Hz noise was completely eliminated by this method.

In medicine they call it Right Leg Guard, RLG, but nowadays in EE we most often refer to this as Right Leg Drive or Driven Right Leg (DRL).